All BMC BAMC 2021 abstracts are listed alphabetically by surname. This is available to download here.
Details of which part of the programme each abstract is associated with is provided after each listing.
Candy Abboud (Glasgow)
Model & data-based Prediction of Invasive-pathogen Dynamics
Prediction of invasive-pathogen dynamics is an essential step towards the assessment of eradication and containment strategies. Such predictions are performed using surveillance data and models grounded on partial differential equations (PDE), which form a framework often exploited to design invasion models. The framework allows the construction of phenomenological but concise models relying on mechanistic hypotheses. However, this may lead to models with overly rigid behaviour, in particular for describing phenomena in population biology. Hence, to avoid drawing a prediction relying on a single PDE-based model that would be prone to errors because of potential data-model mismatch, we propose to apply Bayesian model-averaging (BMA) for handling parameter and model uncertainties. Hence, we combine several competing spatio-temporal models of propagation for inferring parameters and drawing a consensual prediction of certain quantities of interest.
This study is applied (i) to date and localize the invasion of Xylella fastidiosa, bacterium detected in Southern Corsica in 2015, France using post-introduction data, and (ii) to predict its future extent. Work done jointly with Samuel Soubeyrand (INRA, France) and Eric Parent (AgroParisTech, France).
MS13 Mathematical challenges in spatial ecology
Sophie Abrahams (Oxford)
Modelling of laser-induced cavitation bubbles for uteterocopy
We study here the vapour bubbles produced during ‘ureteroscopy and laser lithotripsy’ treatment of kidney stones. This treatment involves passing a flexible ureteroscope containing a laser fibre via the ureter and bladder and into the kidney. The laser fibre is placed in contact with the stone and pulses are fired to fragment the stone into pieces small enough to pass through an outflow channel. Laser energy transferred to the surrounding fluid results in vapourisation between the fibre and stone and the production of a cavitation bubble. While in some cases, bubbles have undesirable effects – for example, causing retropulsion of the kidney stone – it is possible to exploit bubbles to make the fragmentation more efficient. One laser manufacturer employs a method of firing laser pulses in quick succession; the latter pulses pass through the bubble created by the first which increases the energy transferred to the stone. As a first step towards determining the optimal choice of these laser settings, we have developed a model for the behaviour of a single vapour bubble in liquid, as a function of laser energy delivered, pulse duration and pulse pattern. The model employs the Rayleigh-Plesset equation, commonly used in bubble dynamics, and couples it to three equations: the Clausius-Clapyron equation relating the vapour pressure and temperature; a convection-diffusion equation for the surrounding liquid temperature, with a source term for the input of laser energy; and an equation relating the vapour and liquid temperatures. The system is solved numerically for different laser settings – e.g. laser energy, pulse duration and pulse pattern – from which we determine the lifetime and maximum radius of the bubble for each case.
Poster
Mohammed Afsar (Strathclyde)
The role of the spanwise correlation length in low-frequency sound radiation by turbulence/surface interaction
The interaction between non-homogeneous turbulence carried by a non-planar mean field and the discontinuity created by an impermeable solid surface embedded within the flow causes an $O(1)$ increase in the low frequency sound above the background jet noise radiated by the turbulence itself.
When the solid surfaces $S(\boldsymbol{y})$ are placed parallel to the level curves of the streamwise mean flow, $U(y_2,y_3)=const.$, generalized Rapid distortion theory (Ref) shows that the far-field pressure fluctuation, $p^\prime(\boldsymbol{x},t)$, at the spacetime field point $(\boldsymbol{x},t) = ({x}_1,{x}_2,{x}_3,t) $ in a three-dimensional Cartesian co-ordinate system with origin at the trailing edge, can be expressed in terms of the Green's function of the adjoint Rayleigh equation and a convected scalar field, $\tilde{\omega}_c$, defined by $D\tilde{\omega}_c /Dt = 0$ where$D /Dt \equiv \partial/\partial t + U(x_2, x_3) \partial/\partial x_1$ is the material derivative.
Since this latter quantity is an arbitrary function of its arguments, its space-time spectrum ($\bar{S}$ below) can be related to the appropriate measured turbulence correlation function in the upstream undisturbed flow and used as the boundary condition to determine the downstream acoustic field.
In reference 1\footnote{M. E. Goldstein, S. J. Leib \& M. Z. Afsar, {Rapid distortion theory on transversely sheared mean flows of arbitrary cross-section}, \textit{J. Fluid Mech.} (2019), vol. 881, pp. 551–584.}; it is shown that the acoustic spectrum ($I_\omega$) takes the form, $I_\omega \sim \int_u \int_{\tilde {u}} D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega) du d\tilde{u}$, where $D(u,\tilde{u})$ is a directivity factor and $u$ is the real part of a conformal mapping that transforms the nonrectangular cross sectional shape of an axisymmetric jet close to an external surface into a rectangular one for the subsequent Wiener-Hopf technique calculation.
Our main contribution in this paper is to show that the low frequency decay of $I_\omega (\boldsymbol{x})$ is approximately the same as that obtained by taking $l_3 \rightarrow \infty$.
The latter limit allows $\bar{S}$ to take a reduced analytical form. The large-$l_3$ limit corresponds to the allowing the spanwise correlation length to be infinite and therefore that the spectral function $\bar{S}$ is independent of spatial separation in the spanwise (or cross-stream) $3$-direction.
We assess the importance of the spanwise correlation length in controlling the low frequency roll-off of $I_\omega (\boldsymbol{x})$ (i.e. its asymptotic value when the angular frequency, $\omega$, goes to zero starting from the peak sound. Our calculations below show that the maximum value and spatial structure of the integrand, $D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega)$ when $l_3 \rightarrow\infty$ (left figure) is almost identical to the $l_3 = O(1)$ case (right) at very low frequencies.
The accompanying talk will discuss whether the structure of $\bar{S}$ at $l_3\rightarrow \infty$ can be used to approximate $I_\omega$ at more $O(1)$ values of spanwise turbulence length scale, $l_3$.
CT07 High Reynolds Number
Shazia Ahmed (Glasgow)
Spinning a Yarn About Hidden Mathematics
Calling all knitters! Join the University of Glasgow for a live, mathematical-inspired knitting session. Help contribute to our banner for ‘Maths Week Scotland’ by knitting letters and other mathematical symbols.
After signing up, you’ll be posted a free knitting kit including yarn and pattern charts to enable your participation. This event will be delivered online, on Zoom. You will be emailed joining instructions in advance of the event.
Outreach Video
Raneem Aizouk (City, London)
Modelling conflicting individual preference: target sequences and graph realizations
In this presentation we will consider a group of individuals, who each have a target number of contacts they would like to have with other group members. We are interested in how close this can some to being realized, and consider the long term outcome for the group under a reasonable dynamics on the number of contacts. We formulate this as a graph realization problem for undirected graphs, with the individuals as vertices, and the number of contacts as the vertex degree. It is well known that not all degree sequences can be realized as undirected graphs and the Havel-Hakimi algorithm characterizes those that can. When we ask how close the degree sequences can be to realization, we ask for graphs that minimize the total deviation between what is desired and what is possible. The set of all such graphs and the set of all such associated sequences are termed the minimal sets. This problem has previously been considered by Broom and Cannings in a series of papers, and it is a hard problem to tackle for general target sequences. In this talk we consider the n-element arithmetic sequence (n-1, n-2, … 1,0) for general n, including obtaining a formula which generates the size of the minimal set for a given arithmetic sequence. We also consider a strategic version of the model where the individuals are involved in a multiplayer game, with each trying to achieve their target.
Authors: Raneem Aizouk and Mark Broom, City, University of London.
Poster
Tahani Mohammed Sulaiman Al Sariri (Glasgow)
Multiscale modelling of nanoparticle delivery and heat transport in vascularised tumours
Cancer hyperthermia therapies are based on the application of suitable magnetic fields exciting nanoparticles delivered intravascularly. The nanoparticles can interact with the tumour environment via extravasation (the "passive" mechanism), and by binding directly to the tumour receptors (the "active" mechanism). Here, we are interested in (Iron oxide) nanoparticle and temperature maps arising from cancer hyperthermia in vascularised tumours. We start from Darcy’s and Stokes’ problems in the interstitial and fluid vessels compartments, respectively, and then we represent nanoparticle and heat transport via suitable advection-diffusion (heat)-reaction equations in both compartments. We aim at retaining the influence of the micro-vessels on the nanoparticle distribution and temperature maps, however, the system under consideration is intrinsically multiscale. In fact, the distance between adjacent vessels (the microscale) is much smaller than the average tumour size (the macroscale), where experimental measurements are usually performed, thus motivating a homogenization approach. We derive a new system of homogenized partial differential equations which describes blood transport, nanoparticle delivery, and heat by means of the asymptotic homogenization technique. The new model comprises a double Darcy's law for the fluid flow, and a double advection-diffusion (heat)-reaction system for the particle and heat transport (with mass, drug, and heat transport between compartments). The role of the microstructure is encoded in the coefficients of the model which are to be computed solving appropriate microscale periodic problems. According to our numerical results, standard absorption rate associated to magnetic fields that are normally used in cancer hyperthermia therapies can produce a significant (5-6 degrees) temperature increase (which decreases in time due to plasma clearance), which is observed when moving towards the tumour centre from the periphery.
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Nissrin Alachkar (Manchester)
Understanding heterogeneity of immune cell responses through mathematical modelling and analysis of transcriptional bursting
In all living cells, gene expression is a fundamental process that ensures the transfer of genetic information from DNA to RNA to protein which dictates the cell function.
Transcription (DNA to RNA) of most genes is regulated by random transitions between states of gene activity and inactivity, and net messenger RNA (mRNA) production is determined by the frequency and duration of the resulting bursts, causing heterogeneity or “noise” in gene expression at the single cell level.
The aim of this work is to understand how the level of cellular variability is mechanistically regulated. In particular, we measure, using single molecule fluorescence in situ hybridization (smFISH), and mathematically model, the distributions of the number of mRNA molecules for two pro-inflammatory cytokines, Tumour Necrosis Factor alpha (TNFalpha) and Interleukin 1beta (IL1beta ), that play important, but distinct, roles during bacterial infection.
We start modelling stochastic gene expression using two state model; its time evolution is regarded as a random walk process governed by the chemical master equation (CME). Thus, we describe the system with a set of linear ODEs. The probability of finding a cell in each possible state of the system is assigned an ODE and, by solving the resulting system of equations we obtain the time-dependent probability distribution for the model. We also use the Gillespie algorithm to simulate smFISH data.
We show that homogenous expression of TNFalpha is described well by a standard, two-state, stochastic switch model. In contrast, a faithful model of IL1beta transcription requires an additional regulatory step that gives rise to greater heterogeneity via larger bursts sizes and lower bursting frequency. We also show that the gene-by-gene variability is linearly constrained by the mean response across a range of immune-relevant conditions, determining properties of transcriptional bursting that we calculate mathematically.
Poster
Manal Alamoudi (Cardiff)
Stochastic Finite Strain Analysis of Inhomogeneous Hyperelastic Bodies
We examine theoretically the dynamic inflation and finite amplitude oscillations of inhomogeneous cylindrical tubes of stochastic hyperelastic material. We consider composite tubes with two concentric stochastic homogeneous neo-Hookean phases, and inhomogeneous tubes of stochastic neo-Hookean material with constitutive parameters varying continuously in the radial direction. For the homogeneous materials, we define the elastic parameters as spatially-independent random variables, while for the radially inhomogeneous bodies, we take the parameters as spatially-dependent random fields, described by non-Gaussian probability density functions. Under radially symmetric dynamic deformation treated as quasi-equilibrated motion, the bodies oscillate, i.e., the radius increases up to a point, then decreases, then increases again, and so on, and the amplitude and period of the oscillations are characterised by probability distributions, depending on the initial conditions, the geometry, and the probabilistic material properties.
Poster
Hamid Alemi Ardakani (Exeter)
An alternative view on the Bateman-Luke variational principle for wave-body-fluid interactions
A new derivation of the Bernoulli equation for water waves in three-dimensional rotating and translating coordinate systems is given. An alternative view on the Bateman-Luke variational principle is presented. The variational principle recovers the boundary value problem governing the motion of potential water waves in a container undergoing prescribed rigid-body motion in three dimensions. A mathematical theory is presented for the problem of three-dimensional interactions between potential surface waves and a floating structure with interior potential fluid sloshing. The complete set of equations of motion for the exterior gravity-driven water waves, and the exact nonlinear hydrodynamic equations of motion for the linear momentum and angular momentum of the floating rigid-body containing fluid, are derived from a second variational principle.
MS14 Variational Methods in Geophysical Fluid Dynamics
Thoraya Alharthi (Exeter)
Rate-dependent tipping for dynamical system in the the presence of periodic forcing
Rate-induced tipping bifurcation is a tipping mechanism for attractors in certain non- autonomous dynamical systems. Other tipping mechanisms involve bifurcation and noise, but there have been several studies of rate-induced tipping in the presence of parameter shift. The objective of this work is to provide a mathematical framework for rate-induced tipping of certain types of non-autonomous dynamic systems in the presence of periodic forces and to provide the necessary conditions for such behaviour to occur or not. For a specific model system, we examine the rate-induced tipping near a saddle-node bifurcation that is subject to a parameter shift between different types of periodic forcing. In addition, we are investigating the quasi-static behaviour of the system of the rate-induced transitions.
Poster
Jeza Allohibi (Leicester)
New Robust Regression through Huber’s criterion and PQSQ Function
Regression algorithms are almost relying on minimizing or maximizing a function called objective functions. The functions that minimized are called Loss functions. Various loss functions are used to deal with various types of regression-related tasks. In this poster, a series of linear regression models based on different loss functions are performed to analyse the robustness of the parameter coefficients across all selected models. We propose a new robust PQSQ regression model. The proposed method combines the advantages of PQSQ-L1 and PQSQ-L2 regression, which yields the proposed PQSQ-Huber method. The linear regression models used in this study besides the proposed approach are mean square, least absolute deviation, Huber, quantile, MM-estimate, and PQSQ regression. The advantages of the proposed approach over its competitors are demonstrated through both extensive Monte Carlo simulations and twenty real data samples. Both a simulation study and the real data application show that the PQSQ-Huber regression is more robust than its competitors in the presence of outliers.
Poster
Moataz Alosaimi (Leeds)
Identification of the thermo-physical blood-tissue properties
We investigate the retrieval of several thermo-physical blood-tissue properties of a single-layered biological tissue from boundary temperature measurements. The thermal-wave hyperbolic model of bio-heat transfer is used in place of the parabolic diffusive one because in practice there is a non-zero relaxation time between 15 to 30 sec required for a sufficient amount of energy to accumulate and transfer. The retrieval of the thermo-physical properties given by the blood perfusion rate, the thermal contact resistance, the relaxation time, the thermal conductivity and the heat capacity of the tissue from both exact and noisy data is successfully accomplished using a minimisation procedure based on the MATLAB routine lsqnonlin.
Poster
Maram Alossaimi (Sheffield)
Poisson Algebras
A large class of Poisson algebras A is a polynomial Poisson algebra in two variables x and y with coefficients in a polynomial Poisson algebra in one variable t. We classified class A, over an algebraic closure field K with zero characteristic, by using Lemma (Oh, 2006). We concluded that there are three main cases and each case has several subcases. Then we have found the Poisson spectrum of A in each case. In this poster, we identify only the first case.
Poster
Mnerh Alqahtani (Warwick)
Extreme events of Lagrangian model of passive scalar turbulence via large deviation theory
Large deviation theory is the theory behind quantifying the probability of rare and extreme events. These are of interest to physicists, actuaries, biologists, etc., depending on the underlying system. If a large deviation principle (LDP) holds, then the probability of these tail events decays exponentially, but the dominating contribution can be estimated from the minima of the rate function. This poster presents a difficulty of probing these unlikely events in a stochastic differential equation, when the quantity of interest has a heavy-tailed distribution, meaning its rate function is nonconvex. In this case, the standard procedure, which is exponential tilting, fails. We offer a solution, which is a nonlinear reparameterization, justified by convex analysis and the Gärtner-Ellis theorem, including the duality between the cumulant generating function (CGF) and the rate function. We demonstrate the applicability of our method by considering a Lagrangian model of passive scalar turbulence, which exhibits heavy-tailed distribution.
Poster
Yasser Alrashedi (Exeter)
The Effectiveness of Consensus in the Control on Invasive Pests
Urban life faces numerous threats from biological invasions – whether from plants, birds, insects, mammals or pests. Biological invasions can be managed via the application of pes- ticides. However, biological invasions, by their very nature, are often novel and so accurate models for design of management strategies may not be available at all or at best be highly uncertain. Recently, ideas from adaptive control theory have been introduced in the context of managing biological invasions. Several controls strategies and adaptation mecha- nisms were introduced. Here we further develop this approach. Specifically, we are interested in spatially distributed biological invasions. A common strategy that we may consider in order to control pests is to share information about the pests’ location and abundance. Nevertheless, this strategy, depending on the spatial characteristics and mobility of pests, sharing informa- tion consensus, can either hinder or improve individual success. A simple idea of consensus control goes as follows: Suppose there are two farmers – Farmer A and B. Farmer A has an outbreak of pests eating crops and starts applying pesticide control strategies. Farmer B, concerned about a potential spread, asks Farmer A how they dealt with the outbreak. Here we explore the influence of rate of spread of pests vs. the rate of spread of information on the effectiveness of the control strategy.
Poster
Faisal Alsharif (Leicester)
Multilevel quasi-interpolation with the Gaussian using Chebyshev node and edges adjustments.
A review of the numerical methods used today shows that new techniques are required to satisfy the accuracy, convergence, and error estimation of these methods in an approximation view. In this poster, we introduce quasiinterpolation method with the Gaussian via combination technique. Firstly, we will use equally spaced and Chebyshev node in interval x \in [1,1] and calculate the absolute error estimate where we have high absolute error value on the edges. To address this problem, we integrate outside the edges using constant and linear adjustments for both sides to reduce the absolute error value on the edges. We will then compare quasiinterpolation using equally spaced and Chebyshev node with adjustments to show a significant reduction and fast approach to the optimum absolute error value. Finally, we will show some numerical experiment for this funding with different functions and with one level of multilevel quasiinterpolation.
Poster
Mohammed Alsubhi (Exeter)
Cyber-Natural Control systems
Natural populations evolve in response to selection pressures, control systems adapt to prevailing situations. In pest control or anti-biotic treatments of disease these two processes interact: pest control modifies the selection landscape in turn driving the resistance in pest; bacteria may evolve anti-microbial resistance. This interaction of natural evolution and technological adaptation results in a class of ``Cyber Natural Control Systems’’. In this poster, we describe recent results on the dynamics of cyber natural systems. We focus on a class of switched systems. Here the natural system switches between several `phenotypes’. This switched system is then subjected to switched feedback control. For example, a pest may express two or more phenotypes whilst the pest control may have access to two or more pesticides – but neither knows what the other is doing and only accesses each other's responses to the action. The question is what types of closed-loop dynamics emerge. A key driver of these dynamics is the relative rate of switching leading to cycles of rapid growth, slow decline or slow growth rapid decline.
Poster
Lamia Alyami (Exeter)
Comparison of State Estimation Performance of Nonlinear COVID-19 Model using Extended Kalman Filters: Case Study from India and USA
The world health organization(WHO) considers the COVID-19 as a global pandemic that has affected all countries. In response to this outbreak, in this poster, we propose using the recursive estimator called extended Kalman filter with higher-order nonlinear models to analyse the effect of COVID-19 pandemic model and estimate the future states. The susceptible-exposed-infected-quarantined-recovered-dead (SEIQRD) compartmental model has been used to predict the outbreak evolution in the USA and India with daily measurements to understand the infection dynamics for COVID-19. We show here how the Kalman filter algorithm works for this continues time nonlinear system and we compare between three different types of extended Kalman filters which are the hybrid extended Kalman filter, iterated extended Kalman filter, and the second-order Kalman filter. In addition, we examine the potential effectiveness of the precaution measures which are taken by the governments in the form of lockdown rates. Simulation results confirm that the higher order extended Kalman filter lead to better state estimation results to track the growth of the pandemic, in terms of lower estimation error.
Poster
Carlos Amendola (Munich)
Conditional Independence in Max-linear Bayesian Networks
Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. We also introduce the notion of an impact graph which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry. Joint work with Claudia Klüppelberg, Steffen Lauritzen and Ngoc Tran.
MS07 Applied Algebra and Geometry
Cristina Ana Maria Anghel (Oxford)
Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces
The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of link invariants. In this context, the quantum group Uq(sl(2)) leads to the sequence of coloured Jones polynomials, which contains the original Jones polynomial. Dually, the quantum group at roots of unity gives the sequence of coloured Alexander polynomials. On the topological side, Lawrence defined representations of braid groups on the homology of coverings of configurations spaces. Then, Bigelow and Lawrence gave a topological model for the original Jones polynomial.
We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum (defined over 3 variables) of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.
BMC03 Topology
Renato Andrade (Glasgow)
Persistence of intraguild predation in a 1D finite domain using variational approximations
Fragmentation of natural landscapes due to human activity has an undeniable effect on wildlife. A well-known example being deforestation that results in loss of habitat and the subsequent extinction of forest inhabiting species. In this context, a natural question to ask is: what is the minimum habitat size that allows for (co)existence of the species living in a given environment? The work described here seeks to address this for the particular case of three ecologically interacting species: a prey, its predator and a common resource. A system known as an intraguild predation module. We propose and study a model consisting of 3 coupled reaction-diffusion equations. In addition to numerical simulations of the partial differential equations, we approximate the model solution using a method based on variational principles that enable us to derive analytical estimates for critical habitat sizes required for the coexistence of the three interacting species.
MS13 Mathematical challenges in spatial ecology
Andrew John Archer (Loughborough)
Coupled dynamics of premelting films, water droplets, and ice crystal growth
In exciting recent studies of the surface of ice at supersaturation in the vicinity of the triple point, subnanometer height crystal terraces spreading across the surface were observed simultaneously to the formation of micron size droplets, and the subsequent emergence of nanometer thick films below the drops [Murata et al. PNAS 113, E6741 (2016)]. We develop a mesoscopic crystal growth model that takes as input the equilibrium premelting film thickness and interfacial free energies obtained in computer simulations and bridges all relevant scales from angstroms to tens of nanometers. We find that the average dynamics of this complex out of equilibrium system can be described by an effective free energy leading to a wetting phenomenology on the growing ice substrate in complete analogy with equilibrium wetting properties on an inert substrate. Our results explain the significance of the kinetic transition lines observed in the experiments, showing that the motion of the underlying solid surface can be conveyed through the premelted liquid-like layer to the outer surface. Moreover, spontaneous fluctuations in the solid surface can nucleate subsequent deterministic growth of liquid droplets.
CT14 Droplets
Aleksandra Ardaseva (Copenhagen)
A nonlocal PDE model for evolutionary adaptation of cancer cells to fluctuating oxygen levels
We present a system of nonlocal partial differential equations modelling the evolutionary dynamics of phenotype-structured cancer cell populations exposed to fluctuating oxygen levels. In this model, the phenotypic state of every cell is described by a continuous variable that provides a simple representation of its metabolic phenotype. The cells are grouped into two competing populations that undergo heritable, spontaneous, phenotypic variations at different rates. We investigate the development of different environmental niches and explore the adaptation strategies that are selected. These results shed light on the evolutionary processes that may underpin the emergence of phenotypic heterogeneity in vascularised tumours, and suggest potential therapeutic strategies.
MS12 Front Propagation in PDE, probability and applications
Peter Ashwin (Exeter)
Plant ER networks and the dynamics of anchored 2D foams subject to viscous flow
The Endoplasmic Reticulum in plant cells can form a variety of rapidly changing structures including networks of filaments that are anchored to the cell membrane at various points. We discuss progress in biophysical modelling of the interaction of these geometric networks with other processes in play within the cell, in particular actin-driven cross-connections and viscous flow associated with cytoplasmic streaming. We show these processes can maintain an anchored 2D foam of filaments and, maybe more surprisingly the foam retains memory of past streaming speed and direction. (Joint work with Congping Lin, Wuhan).
MS09 Integrating dynamical systems with data driven methods
Mehsin Jabel Atteya (Leicester)
Symmetric Skew n-Antisemigeneralized Semiderivation of (σ,τ)-Anticommutative Rings
The study of derivation was initiated during the 1950s and 1960s. Derivations of rings got a tremendous development in 1957 when Posner [1] established two very striking results in the case of prime rings. J. Bergen [2] introduced the notion of semiderivations of a ring R which extends the notion of the derivation of a ring R, as follows: d: R ⟶ R is a semiderivation of R if there exists a function g: R ⟶ R such that (i) d(xy) = d(x) g(y) + xd(y) = d(x)y + g(x)d(y) for all x, y ∈ R and (ii) d(g(x)) = g(d(x)) for all x ∈ R .
In 2020, Mehsin Jabel Atteya [4] introduced the definition of (σ,τ)-Homgeneralized derivations of semiprime rings with some results as follows: let R be a ring and σ ,τ be automorphism mappings of R . An additive mapping H: R ⟶ R is called a (σ,τ)-Homogeneralized derivation of R if H(xy)=H(x)H(y)+H(x)σ(y)+τ(x)h(y) , where h: R ⟶ R is (σ,τ) -Homoderivation of R for all x,y ∈ R.
The main purpose of this paper is to introduce the definition of (σ,τ)-anticommutative rings. Furthermore, we employing the symmetric skew n-antisemigeneralized semiderivation of (σ,τ)-anticommutative rings. This article divided into two sections, in the first section, we emphasize on the definition of (σ,τ)-anticommutative rings while in the second section, we study the symmetric skew n-antisemigeneralized semiderivation of (σ,τ)-anticommutative prime rings and (σ,τ)-anticommutative semiprime rings. Examples of various results have also been included.
References
[1] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093- 1100.
[2] J. Bergen, Derivations in prime rings}, Canad. Math. Bull., Vol. 26 (3), (1983), 267-270.
[3] Mehsin Jabel Atteya, (σ,τ)-Homgeneralized Derivations of Semiprime Rings, 13th Annual Binghamton University Graduate Conference in Algebra and Topology (BUGCAT), The State University of New York, USA, November 7- 8, and November 14-15, (2020).
Poster
Kamyar Azizzadenesheli (Purdue)
A Crash Course on Neural Operators
Neural Operators are a new advancement in machine learning, applied mathematics, and science, that allows for efficiently learning operators from infinite-dimensional spaces, e.g. function spaces. In this talk, we cover the basics of Neural Operators, their properties, architectures, computation powers, limitations, and the theory behind them. We concluded the talk with a few empirical study partial differential equations (PDEs) to elaborate on the broad applicability of these methods.
MS09 Integrating dynamical systems with data driven methods
Andrew Baggaley (Newcastle)
Classical and quantum vortex leapfrogging in two-dimensional channels
The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterises these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics are richer than in an unbounded domain: alongside the known regimes of standard leapfrogging and the absence of it, we identify new regimes of image-driven leapfrogging and periodic orbits. Moreover, by solving the Gross-Pitaevskii equation for a Bose-Einstein condensate, we show that all four regimes exist for quantum vortices too. Finally, we discuss the differences between classical and quantum vortex leapfrogging which appear when the quantum healing length becomes significant compared to the vortex separation or the channel size, and when, due to high velocity, compressibility effects in the condensate becomes significant.
CT07 High Reynolds Number
Rafael Bailo (Lille)
Projective and Telescopic Projective Integration for Kinetic Mixtures
We propose fully explicit projective integration schemes for non-linear collisional kinetic equations for mixtures. The methods employ a sequence of small forward-Euler steps, intercalated with extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the scheme against standard test cases, and we demonstrate its prowess in dealing with large mass ratios and other complex phenomena. This is joint work with T. Rey.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Scott Balchin (MPIM Bonn)
Equivariant homotopy commutativity and the Catalan numbers
Studying the concept of commutativity up to homotopy is already a difficult problem, however, with the introduction of a group structure the problem is drastically more complicated. Luckily, we shall see that in the case of a finite group, the possible options for equivariant homotopy commutativity can be encoded using simple combinatorics via objects called indexing systems.
In particular, we show that for cyclic groups of prime power order, this combinatorial problem retrieves the construction of the Catalan numbers. We furthermore see a relationship to the associahedra when equipping these indexing systems with a natural order.
[This is joint work with David Barnes and Constanze Roitzheim].
BMC03 Topology
Saleh Baqer (Edinburgh)
Nematic dispersive shocks: low to high optical power
Linearly polarised optical beams with step intensity distributions propagating through azo-doped nematic liquid crystals generate dispersive analogues of classical shock waves. Such wave forms are termed dispersive shocks, or undular bores as known in the context of fluids. This talk starts with an overview of previous work pertaining to dispersive shocks formed at low powers. We then investigate the nature of nematic dispersive shocks when the light wave power is varied from low to high. It is found that the nematic shock structures depend fundamentally upon the magnitude of the input optical power. Furthermore, these optical shocks are resonant with diffractive radiation and the structure of this resonant radiation is also critically dependent upon the power of the beam. Analytical methods based on Whitham’s averaging theory and asymptotic techniques are used to determine solutions for the distinct dispersive shock regimes with the existence intervals for each identified. The analytical solutions obtained are verified via numerical solutions of the nematic equations and excellent agreement is found.
MS23 Dispersive hydrodynamics and applications
Agnese Barbensi (Oxford)
A topological selection of knot folding pathways from native states
A small percentage of catalogued proteins is known to form open ended knots. Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of their projections using topological objects called knotoids. We define a topologically inspired distance between the knotoid distributions of knotted proteins, and we use it to detect and distinguish specific geometrical features for proteins sharing the same dominant topology. Our method allows us to reveal different folding pathways for proteins forming open ended trefoil knots by directly looking at the geometry and topology of their native states. This is joint work with N.Yerolemou, O.Vipond, B.Mahler and D.Goundaroulis.
BMC03 Topology
Ricardo Barros (Loughborough)
A High-order unidirectional model for large-amplitude long internal waves
To describe large amplitude internal solitary waves in a two-layer system, we consider a high-order unidirectional (HOU) model that extends the Korteweg-de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model proposed by Choi and Camassa (1999) is valid only for weakly nonlinear waves, its coefficients depending on the depth and density ratios are adjusted such that the adjusted model can represent the main characteristics of large amplitude internal solitary waves, including effective wavelength, wave speed, and maximum wave amplitude. It is shown that the solitary wave solution of the adjusted HOU (aHOU) model agrees well with that of the strongly nonlinear Miyata-Choi-Camassa (MCC) model up to the maximum wave amplitude, which cannot be achieved by the original HOU model. Numerical solutions of the aHOU model are presented, and it is found that the aHOU model is a simple, but reliable theoretical model for large amplitude internal solitary waves, which would be useful for practical applications.
Reference: Choi and Camassa, J. Fluid Mech. (1999), vol. 396, pp.1-36.
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Aleix Bassolas (QMUL)
First-passage times to quantify spatial heterogeneity: A tale of urban segregation
Virtually all the emergent properties of a complex system are rooted in the non-homogeneous nature of the behaviours of its elements and of the interactions among them. However, the fact that heterogeneity and correlations can appear simultaneously at local, mesoscopic, and global scales, is a concrete challenge for any systematic approach to quantify them in systems of different types. We develop here a scalable and non-parametric framework to characterise the presence of heterogeneity and correlations in spatial systems, based on the statistics of random walks over the underlying network of interactions among its units. In particular, we focus on normalised mean first passage times between meaningful pre-assigned classes of nodes and their application to urban segregation. We introduce the concept of dynamic segregation, that is the extent to which a given group of people, characterized by a given income or ethnicity, is internally clustered or exposed to other groups as a result of mobility and test it in several US cities. While the dynamic segregation of African American communities is significantly associated with the weekly excess COVID-19 incidence and mortality in those communities, the income segregation appears to shape the incidence of socio-economic inequalities.
MS08 Spatial Networks
Stanislaw Biber (Bristol)
Curious dynamics of a Golf Ball bounce
The bounce of a ball in sports such as tennis, cricket or football has been studied extensively with many experimental data available to support the analysis. The common denominator for these models is an impact of a compliant ball off a rigid ground. A bounce of a golf ball is a very different problem though, where the analysis focuses on the impact of a rigid body off a compliant surface. Previous studies of this problem have been largely limited by the lack of experimental data [1]. In our poster we present our approaches at modelling the bounce of a golf ball where we try to match the physical intuition with experimental data. We extend previous work by creating large experimental campaigns. In the modelling we carefully distinguish the slipping and rolling (sliding) scenarios and we approach the problem with a wide spectrum of initial conditions in mind. We show that the obtained data is well correlated using linear models, though divergent from the classical coefficient of restitution approaches. We present a bifurcation analysis of piecewisesmooth Filippov systems leading us to conclusion that the bouncing golf ball can undergo a grazingsliding bifurcation which justifies the use of locally linear piecewisesmooth dynamical models.
[1] Cross, R. (2018). Backward bounce of a spinning ball. {\it Eur. J. of Phys.}, 39(4), 045007.
Poster
Callum Birkett (Dundee)
Magnetic Helicity and the Calabi Invariant
Authors: C.Birkett, G.Hornig
Magnetic helicity is an important tool in the study of both astrophysical and laboratory plasmas. The topological features of the magnetic field confined to the plasma provide estimates on the energy contained within. We introduce the Calabi invariant, an integral quantity closely associated with helicity (Calabi 1970, Gambaudo et al. 2000) and show that this leads to interesting new ways to interpret the helicity and to calculate it in magnetically open domains. The Calabi invariant allows us to assess the topology of a plasma contained in a domain without having to construct the entire field - we only need the associated field line mapping. An application to braiding within a coronal loop is presented.
References
[1] Eugenio Calabi. On the group of automorphisms of a symplectic manifold, from: “problems in analysis (lectures at the sympos. in honor of salomonbochner, princeton univ., princeton, nj, 1969)”, 1970.
[2] Jean-Marc Gambaudo and Maxime Lagrange. Topological lower bounds on the distance between area preserving diffeomorphisms .Boletim da SociedadeBrasileira de Matematica, 31(1):9–27, 2000.
CT11 Magnetohydrodynamics
Yaw Boakye-Ansah (Strathclyde)
Similarity solutions for early-time infiltration in foams and soils
Richards' equation [1] and the foam drainage equations [2,3] describe transport of moisture within soils and foams respectively during infiltration processes. These transport equations each reduces to a nonlinear diffusion equation in the early stage of infiltration. Nonlinear diffusion equations such as these arise quite generally in the early-time evolution of infiltration processes into porous media such as soils and foams, during which time the system is dominated by capillary-driven drainage. New solutions based on the van Genuchten [4] relative diffusivity function for soils are found at early times, and compared with the early-time solutions of nonlinear diffusion for channel-dominated foam drainage [2] These solutions are found using the principle of self-similarity for a constant rate infiltration process. We obtain singular solutions in the case of soils while the channel-dominated foam drainage solutions go to zero moisture content in a gradual fashion. The solutions obtained for the node-dominated [3] foam drainage are found in literature (the governing equation being now linear is analogous to the linear equation for heat transfer). Similarities and differences between the various solutions for nonlinear and linear diffusion are highlighted.
References
[1] L. A. Richards. Capillary conduction of liquids through porous mediums. Physics, 1(5):318–333, 1931.
[2] M. Th Van Genuchten. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44(5):892–898, 1980.
[3] G. Verbist, D. Weaire, and A. M. Kraynik. The foam drainage equation. Journal of Physics: Condensed Matter, 8(21):3715–3736, 1996.
[4] S. A. Koehler, S. Hilgenfeldt, and H. A. Stone. Liquid flow through aqueous foams: the node-dominated foam drainage equation. Physical review letters, 82(21):4232–4235, 1999
Poster
Onno Bokhove (Leeds)
Variational principle for novel wave-energy device
We have developed a novel wave-energy device as well as its accompanying mathematical and numerical model. The wave-energy device consists of a contraction focussing and amplifying the incoming waves, a wave-activated buoy constrained to move in only one direction (or along an arc) and an electro-magnetic induction motor responsible for the conversion of the energy in the buoy motion to electrical power. The variational principle describing the entire wave-energy device will be introduced as well as additional dissipative elements (such as electrical resistance and loads). This principle consists of the variational principles of the separate hydrodynamical, mechanical and electrical elements, and their coupling. Subtleties in the waterline hydrodynamics will be highlighted, together with the variational structure of the electro-magnetic induction motor.
Reference: Bokhove O, Kalogirou A, Zweers W. 2019. From bore-soliton-splash to a new wave-to-wire wave-energy model. Water Waves Vol 1, issue 2, 217-258. 10.1007/s42286-019-00022-9
MS14 Variational Methods in Geophysical Fluid Dynamics
Oliver George Bond (Oxford)
Mathematical Modelling of Thermoelectric Liquid Lithium Behaviour inside a Tokamak Fusion Reactor
Over the last seven decades, research into magnetohydrodynamic (MHD) duct flow has attracted great interest, particularly with respect to applications to nuclear fusion reactors. More recently, concepts which utilise the thermoelectric effect, thus resulting in thermoelectric magnetohydrodynamics (TEMHD), have been proposed and successfully tested inside tokamak fusion reactors. One such concept is the Liquid Metal-Infused Trenches (LiMIT) from the University of Illinois at Urbana-Champaign, which consists of several parallel stainless steel channels down which liquid lithium is allowed to flow, and the flow is driven by an impinging heat flux from the plasma onto the free surface. Such a concept would then be installed onto a component of a tokamak called a divertor; a large axisymmetric substrate upon which heat and particle matter is deposited. The divertor needs to be able to withstand high heat fluxes similar to those of a space shuttle upon re-entry (around 10 MW/m²).
Tokamak Energy, a fusion company in Didcot, Oxfordshire, are interested in different divertor geometries; in particular, if the divertor plate consists of an array of posts rather than trenches, in order to allow lateral motion of the fluid between channels. This project is interested in the mathematical modelling of the TEMHD flow on such divertors, and preliminary attempts at modelling trench-like flow have resulted in a leading-order problem not dissimilar to classical MHD duct flow models such as that of Shercliff in 1953. We are interested in applying singular perturbation methods to the problems obtained when the Hartmann number is large.
Poster
Christian Bönicke (Glasgow)
Dynamic asymptotic dimension and groupoid homology
Dynamic asymptotic dimension is a dimension theory for group actions and more generally for étale groupoids developed by Guentner, Willett, and Yu, which generalizes Gromov’s theory of asymptotic dimension. Having finite asymptotic dimension is known to have important implications for the structure of the involved C*-algebras. In this talk I will report on recent joint work with Dell’Aiera, Gabe, and Willett in which we prove a homology vanishing result for groupoids with finite dynamic asymptotic dimension. Our result allows us to present the first abstract class of groupoids satisfying Matui’s HK conjecture, which claims that the K-theory of a groupoid C*-algebra is completely encoded in the homology of the groupoid.
BMC04 Operator Algebras
Michael Bonsall (Oxford)
Optimal control approaches to understanding COVID-19 pandemic
SARS-CoV-2 virus has spread rapidly across the world and current measures for controlling spread are focused on the use of non-pharmaceutical interventions (NPIs, such as lockdowns, social distancing, mask wearing etc.) and vaccine roll-outs. In this talk, I will present recent work using optimal control approach to understand strategies to control the disease. We use both numerical and analytical approaches to solve complex optimal control problems (extending mathematical approaches in solving systems with time delays). Our focus will be on how scaling in relaxing NPIs, sharing vaccines, or combining vaccines with NPIs can be used to mitigate infections and provide achievable public health outcomes.
MS05 Multiscale Modelling of Infectious Diseases
Agnieszka Borowska (Glasgow)
Parameter estimation and uncertainty quantification in a stochastic differential equation model of cell movement and chemotaxis
Chemotaxis is a type of cell movement in response to a chemical stimulus which plays a key role in multiple biophysical processes, such as embryogenesis and wound healing, and which is crucial for understanding metastasis in cancer research. In the literature, chemotaxis has been modelled using bio-physical models based on systems of nonlinear stochastic partial differential equations (NSPDEs), which are known to be challenging for statistical inference due to the intractability of the associated likelihood and the high computational costs of their numerical integration. Therefore, data analysis in this context has been limited to comparing predictions from NSPDE models to laboratory data using simple descriptive statistics. We present a statistically rigorous framework for parameter estimation in complex biophysical systems described by NSPDEs such as the one of chemotaxis. We adopt a likelihood-free approach based on approximate Bayesian computations with sequential Monte Carlo (ABC-SMC) which allows for circumventing the intractability of the likelihood. To find informative summary statistics, crucial for the performance of ABC, we propose to use a Gaussian process (GP) regression model. The interpolation provided by the GP regression turns out useful on its own merits: it relatively accurately estimates the parameters of the NSPDE model and allows for uncertainty quantification, at a very low computational cost. We demonstrate that the correction provided by ABC-SMC is essential for accurate estimation of some of the NSPDE model parameters and for more flexible uncertainty quantification. Our proposed methodology was externally assessed at the Cside 2018 competition, where it ranked 1st in the category “stochastic differential equations.
MS04 Stochastic models in biology informed by data
Patrick Bourg (Southampton)
Rotating perfect fluid stars in 2+1 dimensions.
Classical Einstein gravity in 2+1 dimensions may at first appear to be trivial since the Weyl tensor is identically zero. However, this apparent simplicity hides, upon closer examination, interesting features. We will review perhaps one of the simplest axistationary matter solutions: rotating perfect fluid stars. These solutions can be written down explicitly in 2+1 dimensions and we will highlight the similarities and differences to their 3+1 counterparts. If time permits, we will also show how these solutions are relevant to the critical collapse of a perfect fluid.
CT 09 Mathematical physics
Paul Bowen (Exeter)
Consistent Modelling of Non-Equilibrium Thermodynamic Processes in the Atmosphere
Approximations in the moist thermodynamics of atmospheric/weather models are often inconsistent. Different parts of numerical models may handle the thermodynamics in different ways, or the approximations may disagree with the laws of thermodynamics. In order to address these problems, we may derive all relevant thermodynamic quantities from a defined thermodynamic potential; approximations are then instead made to the potential itself — this guarantees self-consistency. This concept is viable for vapor and liquid water mixtures in a moist atmospheric system using the Gibbs function but on extension to include the ice phase an ambiguity presents itself at the triple-point. In order to resolve this the energy function must be utilised instead; constrained maximisation methods can then be used on the entropy in order to solve the system equilibrium state. Once this is done however, a further extension is necessary for atmospheric systems. In the Earth’s atmosphere many important non-equilibrium processes take place; for example, freezing of super-cooled water, evaporation, and precipitation. To fully capture these processes the equilibrium method must be reformulated to involve finite rates of approach towards equilibrium. This may be done using various principles of non-equilibrium thermodynamics, principally Onsager reciprocal relations. A numerical scheme may then be implemented which models competing finite processes in a moist thermodynamic system.
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Rodolfo Brandao (Imperial College)
Spontaneous dynamics of Leidenfrost drops
Recent experiments have revealed that Leidenfrost drops (levitated by their vapour above a hot surface) undergo “symmetry breaking”, leading to spontaneous rolling motion in the absence of external gradients/asymmetries (A. Bouillant et al., Nature Physics, 14 1188, 2018). Motivated by these observations, we theoretically investigate the dynamics of Leidenfrost drops on the basis of a simplified two-dimensional model, focusing on near-circular drops small relative to the capillary length. The model couples the equations of motion of the drop, which flows as a rigid wheel, and instantaneous thin-film equations governing the vapour flow, the profile of the deformable vapour-liquid interface, and thus the hydrodynamic forces and torques on the drop. The model predicts an instability of the symmetric Leidenfrost state, manifested by a supercritical pitchfork bifurcation, which leads to spontaneous motion. Our model illuminates several aspects of the experiments, including the origins of the experimentally measured propulsion force and its dependence on the physical parameters of the system, in compelling qualitative agreement with the experiments.
CT14 Droplets
Vitalijs Brejevs (Glasgow)
Ribbon surfaces for alternating 3-braid closures
We construct ribbon surfaces of Euler characteristic one for several infinite families of alternating 3-braid closures. We also give examples of potentially non-slice alternating 3-braid closures whose double branched covers bound rational balls.
Poster
Georgia Brennan (Oxford)
Mathematically Modelling Clearance in Alzheimer's Disease
Mankind faces an aging crisis with Alzheimer's disease (AD) at the forefront. Clinical literature increasingly points to the connection between AD and the failure of the brain's ability to remove dangerous waste proteins with age. My research focus is to develop an understanding of how clearance deficits can play a formative role in AD. I am developing the first mathematical, data-driven, network models coupling the clearance of toxic proteins and AD progression. Our groups accessible research software enables clinical researchers to harness key mathematical insights for experimental research and treatment, producing simulations for the spread and dynamics of AD over 40 years in a matter of seconds.
Poster
Matthew Bright (Liverpool)
Introduction to Periodic Topology for Textiles and Crystals
A structure embedded in n-dimensional space is k-periodic if it maps to itself under k independent translations. This is a property of interest to materials scientists - textiles form 2-periodic structures, crystals are in general 3-periodic. Topological invariants of k-periodic structures are therefore of interest as a means of investigating and classifying materials, and potentially designing new ones.
We can consider k-periodic structures as the cover of some finite object - such as an embedding in the flat torus (1) or as a labelled graph (2). In all cases this involves selecting a fundamental repeating unit of the structure - a unit cell. Since there is no unique selection of unit cell, the appropriate equivalence relation in this context is periodic isotopy, which encompasses isotopic deformations of the object itself and all possible unit cell selections.
Our recent work has developed and refined tools for investigating periodic isotopy. We have used the presentation of textiles embedded in a thickened torus in to extend a knot representation - the Gauss code (3) - to the 2-periodic setting. This can be used as input to an efficient algorithm for determining the realisability of an arbitrary code as a textile structure (4) In the 3-periodic context we have developed a closed form for the periodic linking number (5) that allows it to be quickly computed for a crystal structure considered as a spatially embedded graph (6).
References
(1) Grishanov, S. et al. Textile Research Journal 79(8):702-713 (2009)
(2) Eon, J-G. Act. Cryst. A. 72: 268-293 (2015)
(3) Kurlin, V. Math. Proc. Cambridge Phil. Soc. 145: 129-140 (2008)
(4) Bright, M. et al. Computers and Graphics 90: 51=61 (2020)
(5) Panagiotou, E. J. Computational Physics 300: 533-573 (2020)
(6) Bright, M. et al. arXiv:2011.04631v2 (2020)
MS11 Mathematics for Materials Science
Chris Budd (Bath)
Mathematical models for the ice ages
The ice ages are significant changes to the Earth's climate. For the last half a million years they have demonstrated a strong regularity, with large periodic changes in temperature with a 100 kilo year cycle. Before then the climate showed smaller smaller periodic changes, with a 40 kilo year cycle. Although there are many theories for this behaviour, there is as yet no fully convincing explanation for them. In this talk I will apply recent methods from the theory of non-smooth dynamical systems to gain some insight into this complex phenomenon. In particular I will ask the question of whether the 'min-Pleistocene transition (MPT)' half a million years ago was an example of a grazing bifurcation.
Joint work with Kgomotso Susan Morupisi
CT13 Dynamical Systems
Jeremy Michael Budd (TU Delft)
A semi-discrete scheme for graph Allen--Cahn as a classification/segmentation algorithm
An emerging technique in clustering, segmentation and classification problems is to consider the dynamics of flows defined on finite graphs. In particular Bertozzi and co-authors considered dynamics related to Allen—Cahn flow (Bertozzi, Flenner, 2012) and the MBO algorithm (Merkurjev, Kostic, Bertozzi, 2013) for this purpose. Previous work by the authors (Budd, Van Gennip, in review, arxiv preprint 1907.10774) devised a "semi-discrete" scheme for Allen--Cahn, of which MBO is a special case.
This talk will extend this earlier theory to the case of Allen--Cahn/MBO with fidelity forcing. We will then explore the prospects of this semi-discrete scheme (with fidelity) as an alternative to MBO for segmentation and classification applications.
CT 15 Statistical and Numerical Methods
Andrew Burbanks (Portsmouth)
Computer-assisted proof of the existence of renormalisation fixed points
We prove the existence of a fixed point to the renormalisation operator for period doubling in maps of even degree at the critical point. Building on previous work, our proof uses rigorous computer-assisted means to bound operations in a space of analytic functions and hence to show that a quasi-Newton operator for the fixed-point problem is a contraction map on a suitable ball.
We bound the spectrum of the derivative of the renormalisation operator at the fixed point, establishing the hyperbolic structure, in which the presence of a single essential expanding eigenvalue explains the universal asymptotically self-similar bifurcation structure observed in the iterations of families of maps with the relevant degree at the critical point.
By recasting the eigenproblem for the Frechet derivative in nonlinear form, we use the contraction mapping principle to gain rigorous bounds on eigenfunctions and their corresponding eigenvalues. In particular, we gain tight bounds on the eigenfunction corresponding to the essential expanding eigenvalue delta.
By employing a recursive scheme based on the fixed-point equation, we exhibit the structure of the domain of analyticity of the renormalisation fixed point.
Our computations use multi-precision arithmetic with rigorous directed rounding modes (conforming to relevant standards IEEE754-2008 and ISO/IEC/IEEE60559:2011) to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.
CT13 Dynamical Systems
Helen Burgess/David Dritschel (St Andrews)
Potential vorticity fronts and the late-time evolution of large-scale quasigeostrophic flow
The long-time behaviour of freely evolving quasigeostrophic flows with initial characteristic length scale L0 larger than or equal to the deformation radius LD, L0/LD ≥ 1, is studied. At late time the flows are dominated by large multilevel vortices consisting of ascending terraces of well-mixed PV, i.e. PV staircases. We study how the number of mixed PV levels depends on the initial conditions, including L0/LD. For sufficiently large values of L0/LD ≈ 5, a complete staircase with regular steps forms, but as L0/LD decreases the staircase becomes more irregular, with fewer mixed levels and the appearance of a large step around q = 0, corresponding to large regions of near-zero PV separating the multilevel vortices. This occurs because weak PV features in the initial field with scale smaller than LD undergo filamentation and are coarse-grained away by contour surgery. For all values of L0/LD considered inverse cascades of potential energy commence at sufficiently late times. The onset of inverse cascades of potential energy even when the flow is initialised well within the ‘asymptotic model’ (AM) regime suggests that the AM regime is not self-consistent: when potential vorticity fronts are well-resolved, frontal dynamics eventually drive ongoing flow evolution.
MS16 Eddies in Geophysical Fluid Dynamics
Kevin Buzzard (Imperial College London)
When will computers prove theorems?
Computers can now beat us at chess and at go. When will they start to beat us at proving theorems? What kind of theorems might they prove, and what might their proofs look like? What is a proof anyway? Do humans prove theorems, or do they just sketch proofs? If the proof of a theorem is "known to the experts" and then one day there are no experts left, is this still a proof? Will humans and computers be able to collaborate in the future? And when will all this happen -- in 5 years or in 50 years? I will survey the state of the art.
BMC Morning Speaker
Diogo Caetano (Warwick)
Well-posedness for the Cahn-Hilliard equation on an evolving surface
The classical Cahn-Hilliard equation is a fourth order, semilinear parabolic PDE which was first proposed in 1958 to describe phase separation in binary alloys, and it has since been applied to problems in other areas such as image processing, tumor growth models, among others. In this talk, we describe a functional framework suitable to the formulation of the constant mobility Cahn-Hilliard equation on an evolving surface and establish well-posedness for general regular potentials, the thermodynamically relevant logarithmic potential and a double obstacle potential. It turns out that, for the singular potentials, conditions on the initial data and the evolution of the surfaces are necessary for global-in-time existence of solutions, which arise from the fact that the integral of solutions are preserved over time. Time permitting, related models, examples and open questions will be discussed.
CT17 Solid Mechanics
Isabelle Cantat (Rennes)
Marginal pinch stability in foam films
The thinning of the liquid films separating bubbles in a foam or in a bubbly liquid controls the coalescence process and the foam stability, and is highly relevant in many industrial processes. The spatio-temporal evolution of the film thickness is governed by highly nonlinear equations, which solution properties are still mostly unknown. For a flat film in contact with a meniscus at a lower pressure, a classical theoretical solution is the growth of a pinch, invariant along the meniscus. However, film thinning has been shown to be mainly controlled by the non-invariant marginal regeneration for which no clear explanation has been provided to date. We establish experimentally that the theoretical invariant pinch is unstable, even in a horizontal film, and we measure the instability wavelength along the meniscus. We show that the Poiseuille flow can be neglected during the destabilization process, and, taking advantage of this scale separation, we built an original equation set leading to the prediction of the instability wavelength and growth rate.
MS24 The mathematics of gas-liquid foams
Claudio Capelli (UCL)
Translating computational modelling to clinics: opportunities and challenges in the field of congenital heart diseases
Congenital disease are structural and functional defects developed during prenatal life and that remain present after birth. An estimated 1% of children are born worldwide with congenital heart disease. With modern advances in medicine most of the structural congenital anomalies can be corrected thanks to the massive progress made in paediatric surgical procedures, interventions and medical technologies. Despite this, life expectancy for children with critical congenital abnormalities is lower than the average population and, in some cases, patients require continued treatment throughout the patient’s life. The complexity of these conditions often require personalized and tailored approaches. In this context, in silico medicine including structural and fluid-dynamic analyses can indeed provide support to enrich diagnosis and model personalized treatments for children born with congenital diseases. In his talk, Claudio Capelli will speak about his experience as a bioengineering researcher embedded within a clinical centre of excellence (i.e. the Great Ormond Street Hospital for Children). Methods and results will be presented highlighting both successes and challenges incurred in over a decade of efforts to translate computational models and simulation to clinics.
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Magda Carr (Newcastle)
Shoaling Mode-2 Internal Solitary-Like Waves
Internal solitary waves (ISWs) propagate along density interfaces in stably-stratified fluid systems. They occur frequently in geophysical settings such as estuaries, lakes, fjords, oceans, marginal seas and the atmosphere. They owe their existence to a balance between nonlinear wave steepening and linear wave dispersion. In linear theory, sets of modal solutions exist for large amplitude ISWs propagating in bounded, stratified fluids but, for most cases of geophysical interest, over 90% of kinetic energy of the nonlinear baroclinic modes is contained within the first two modes. Mode-1 ISWs displace isopycnals in one direction only and can be waves of depression or elevation. Mode-2 ISWs on the other hand, displace isopycnals in opposite directions and can be convex or concave in form.
Laboratory investigation and numerical simulation of the propagation of mode-2 ISWs over a uniformly sloping, solid topographic boundary, will be presented. The waves are generated by a lock-release method. Features of their shoaling include (i) formation of an oscillatory tail, (ii) degeneration of the wave form, (iii) wave run up, (iv) boundary layer separation, (v) vortex formation and re-suspension at the bed and (vi) a reflected wave signal. In shallow slope cases, the wave form is destroyed by the shoaling process; the leading mode-2 ISW degenerates into a train of mode-1 waves of elevation and little boundary layer activity is seen. For steeper slopes, boundary layer separation, vortex formation and re-suspension at the bed are observed. The boundary layer dynamics is shown (numerically) to be dependent on the Reynolds number of the flow.
Reference: Carr et al. J. Fluid Mech. (2019), vol. 879, pp. 604632
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Jose Antonio Carrillo (Oxford)
Nonlocal Aggregation-Diffusion Equations: entropies, gradient flows, phase transitions and applications
This talk will be devoted to an overview of recent results understanding the bifurcation analysis of nonlinear Fokker-Planck equations arising in a myriad of applications such as consensus formation, optimization, granular media, swarming behavior, opinion dynamics and financial mathematics to name a few. We will present several results related to localized Cucker-Smale orientation dynamics, McKean-Vlasov equations, and nonlinear diffusion Keller-Segel type models in several settings. We will show the existence of continuous or discontinuous phase transitions on the torus under suitable assumptions on the Fourier modes of the interaction potential. The analysis is based on linear stability in the right functional space associated to the regularity of the problem at hand. While in the case of linear diffusion, one can work in the L2 framework, nonlinear diffusion needs the stronger Linfty topology to proceed with the analysis based on Crandall-Rabinowitz bifurcation analysis applied to the variation of the entropy functional. Explicit examples show that the global bifurcation branches can be very complicated. Stability of the solutions will be discussed based on numerical simulations with fully explicit energy decaying finite volume schemes specifically tailored to the gradient flow structure of these problems. The theoretical analysis of the asymptotic stability of the different branches of solutions is a challenging open problem. This overview talk is based on several works in collaboration with R. Bailo, A. Barbaro, J. A. Canizo, X. Chen, P. Degond, R. Gvalani, J. Hu, G. Pavliotis, A. Schlichting, Q. Wang, Z. Wang, and L. Zhang. This research has been funded by EPSRC EP/P031587/1 and ERC Advanced Grant Nonlocal-CPD 883363.
QJMAM Plenary
Matteo Casati (Ningbo)
Discrete Poisson Cohomology
I will present an unified formalism to describe functional Poisson bivectors in the differential and differential-difference case.
This allows to define their associated Poisson-Lichnerowicz cohomology, carrying information about the Casimir functionals, the symmetries and the admissible deformations of the corresponding Hamiltonian operator. The notion has been widely investigated in the differential case: here I will present the results for (-1,1) order scalar difference operators. An extension of the notion to the noncommutative cases will be briefly discussed to. (partly based on Casati, Wang, "A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators", Commun. Math. Phys. (2019)
BMC05 Math. Physics
Daniele Celoria (Oxford)
A discrete Morse perspective on knot projections
We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse matching (dMm) on the 2-sphere, extending a construction due to Cohen. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMms, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of simple moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations. This is joint work with Naya Yerolemou.
BMC03 Topology
Marianna Cerasuolo (Portsmouth)
On the role of diffusion on drug interaction in the treatment of prostate cancer.
A hybrid system of ODEs and PDEs has been implemented to assess the role of cells and chemicals diffusion on the dynamics of prostate cancer in a multistage murine model TRAMP (transgenic adenocarcinoma of the mouse prostate) under different therapeutic strategies. The model describes the interdependence of cancer cells on tumour microenvironment as well as the onset of resistance following treatment with a second generation drug (androgen receptor antagonist) called enzalutamide.
The proposed mathematical model, whose development strongly relied on experimental data and their statistical analysis, represents a theoretical framework to bridge the in vitro and in vivo experiments used to assess the effect of single- or combined-drug therapies on TRAMP mice and TRAMP-derived cells. The model revealed that combination therapies can delay the onset of resistance to enzalutamide, and in the suitable scenario with alternating drug therapies, can eliminate the disease. The model also showed that some of the drug combinations can cause the formation of smaller-size tumour clusters, which could give rise to metastasis.
CT10 Mathematical Biology-2
Dwaipayan Chakrabarti (Birmingham)
Engineering Open Crystals: Programming Voids with Designer Colloids
Colloidal particles offering anisotropic interactions appeal as “designer” building blocks for programmable self-assembly of soft materials [1,2]. Colloidal open crystals, which are sparsely populated periodic structures, having a maximum density lower than what can be achieved at closepacking, have been attractive targets for programmed self-assembly for their variety of applications as photonic crystals, phononic and mechanical metamaterials, as well as porous media [3-6]. However, programming self-assembly of colloidal particles into open crystals has proved elusive for the challenges that arise from thermodynamic as well as kinetic considerations. In this presentation, building on our earlier work [7], I will establish a hierarchical self-assembly scheme for triblock patchy particles and show its versatility in order to address these challenges while hewing closely to current experimental constraints [8-10]. The presentation will demonstrate in silico the hierarchical self-assembly of colloidal open crystals via the so-called “colloidal molecules” – small colloidal clusters mimicking the symmetry of molecules. By employing a variety of computer simulation techniques, I will show that the design space supports distinct colloidal molecules (e.g. tetrahedra or octahedra with variable valences) en route to a number of colloidal open crystals. In particular, the versatile design framework opens up the prospects for realising certain colloidal open crystals much sough-after for their attractive photonic applications. I will discuss how our design rules alleviate issues that have impeded the success of colloidal self-assembly as a scalable fabrication route to photonic crystals [10].
References
[1] S. C. Glotzer and M. J. Solomon, Nature Mater. 6, 557 (2007).
[2] L. Cademartiri and K. J. M. Bishop, Nature Mater. 14, 2 (2015).
[3] X. Mao, Q. Chen and S. Granick, Nature Mater. 12, 217 (2013).
[4] J. D. Joannopoulos, P. R. Villeneuve and S. Fan, Nature 386, 143 (1997).
[5] K. Aryana and M. B. Zanjani, J. Appl. Phys. 123, 185103 (2018).
[6] X. Mao and T. C. Lubensky, Annu. Rev. Condens. Matter Phys. 9, 413 (2018).
[7] D. Morphew and D. Chakrabarti, Nanoscale 10, 13875 (2018).
[8] D. Morphew, J. Shaw, C. Avins and D. Chakrabarti, ACS Nano 12, 2355 (2018).
[9] A. B. Rao, J. Shaw, A. Neophytou, D. Morphew, F. Sciortino, R. L. Johnston and D. Chakrabarti, ACS Nano 14, 5348 (2020).
[10] A. Neophytou, V. N. Manoharan and D. Chakrabarti, ACS Nano DOI:10.1021/acsnano.0c07824 (2021).
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Robert Chamberlain (Warwick)
Minimal Permutation Representations of Finite Groups
Permutation representations are commonly used to represent finite groups on a computer. The degree of such representations can significantly effect the speed of computations involving the group. It is therefore ideal to use a faithful permutation representation of least degree. Computing this in general is not currently feasible, but this talk provides some of the more useful basic results and some new results, including the minimal degree of a faithful permutation representation of $2.A_n$, the double cover of the alternating group $A_n$, for each $n$.
BMC09 Groups
Long Chen (Durham)
The topology of resistive magnetic relaxation
In plasmas, complex braided magnetic structures are known to self-organise into simple configurations.
While Parker (1983) has conjectured that reconnection events will ultimately lead to only two flux tubes with opposite helicity, Yeates et al. (2010, 2015) have identified the topological degree as a separate constraint. We trace the evolution of topology using a combination of 3D and 2D simulations. We find that there are two distinct phases: a fast reconnection phase constrained by the topological degree, followed by a diffusion dominated phase with the merging of discrete flux tubes. Resistivity and boundary conditions both affect the reconnection events and the resulting topology. Whether the final state could reach Parker’s state depends on various aspects, which I will discuss in this talk.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
Katie Chicot (Maths World UK)
Maths World UK
Maths World UK is aiming to establish the UK's first National Mathematics Discovery Centre.
There is at present no dedicated place in the United Kingdom where you can go to experience the full joy, wonder and power of mathematics.
The Discovery Centre will excite the imagination by encouraging interaction with mathematical objects and images. These will stimulate mathematical thinking in unusual ways that make it possible to appreciate the beauty, ingenuity, applicability and importance of Mathematics. It will include fun problems and puzzles to engage its full range of visitors. View our short video making the case for the Discovery Centre.
Further information: www.mathsworlduk.com
Outreach Video
Yemon Choi (Lancaster)
Completely almost periodic elements of group von Neumann algebras
On a locally compact group $G$, one may consider those $h\in L^\infty(G)$ whose left (or right) translates form a relatively compact subset of $L^\infty(G)$ in the norm topology: such functions are said to be (Bochner-)almost periodic, and they form a unital $C^{\ast}$-subalgebra ${\rm AP}(G)\subseteq L^\infty(G)$, whose Gelfand spectrum coincides with the Bohr compactification of $G$. In particular, if $G$ is compact then ${\rm AP}(G)=C(G)$.
There is a natural analogue of this construction where $L^\infty(G)$ is replaced by ${\rm VN}(\Gamma)$ for a locally compact group $\Gamma$, and the action of the algebra $L^1(G)$ on $L^\infty(G)$ is replaced by the action of the Fourier algebra ${\rm A}(\Gamma)$ on ${\rm VN}(\Gamma)$. Operator space considerations suggest that we should replace the usual notion of compactness by one which takes into account matricial structure, and the resulting space ${\rm CAP}(\widehat{\Gamma})$ is the subject of this talk. We will sketch a proof that ${\rm CAP}(\widehat{\Gamma})$ is always a unital $C^{\ast}$-subalgebra of ${\rm VN}(\Gamma)$, significantly extending previous results of Runde who established this under amenability/injectivity assumptions, and we will indicate how the question ``is ${\rm CAP}(\widehat{\Gamma})$ equal to $C^{\ast}_r(\Gamma)$ whenever $\Gamma$ is discrete?'' is equivalent to an open problem concerning the uniform Roe algebra.
BMC04 Operator Algebras
Avni Chotai (Imperial College)
The effect of wall compliance upon the stability of annular Poiseuille-Couette flow
Driven by a constant axial pressure gradient, the viscous flow through the concentric annular region between a stationary outer cylinder and a sliding inner cylinder is known as annular Poiseuille-Couette flow (APCF). This flow is of particular relevance in medicine; thread-injection is a minimally invasive technique for the transportation of medical implants into the body. In view of this application, the inner cylinder is modelled as compliant in our study. We consider steady, incompressible APCF between two infinitely long, concentric cylinders. The linear stability of this flow to infinitesimal, axisymmetric disturbances is studied asymptotically at large Reynolds numbers and computationally at finite Reynolds numbers when the inner cylinder possesses a degree of flexibility. Typical no-slip conditions apply on the wall of the outer rigid cylinder, and the inner cylinder is described by a spring-backed plate model.
At finite Reynolds number, our problem forms a generalised eigenvalue problem that can be solved numerically via a Chebyshev collocation method to obtain the wavenumbers of the neutral modes. In addition to showing classical viscous modes, the results show that the compliant nature of the inner cylinder results in the existence of unstable ‘elastic’ modes that are not present in the rigid counterpart of the problem.
Both the viscous and elastic modes can be described using asymptotic analysis, with the viscous modes assuming a multizone structure typical of shear flow instability. A comparison of the asymptotic solutions with the finite-Reynolds number solutions shows increasing agreement at larger Reynolds numbers.
Authors - Avni Chotai, Andrew Walton
CT07 High Reynolds Number
Marina Chugunova (Claremont)
Mathematical modeling of pressure regimes in Fontan blood flow circulation
Babies born with only a single functioning heart ventricle require a series of surgeries during their first few years of life to redirect their blood flow. The resulting circulation, in which systemic venous blood bypasses the heart and flows directly into the pulmonary arteries, is referred to as Fontan circulation. We develop two mathematical lumped-parameter models for blood pressure distribution in Fontan blood flow circulation: an ODE based spatially homogeneous model and a PDE based spatially inhomogeneous model. Numerical simulations of the ODE model with physiologically consistent input parameters and cardiac cycle pressure-volume outputs reveal the existence of a critical value for pulmonary resistance above which cardiac output dramatically decreases.
Joint work with M.G. Doyle, J.P. Keener, and R.M. Taranets
MS27 Mathematical and Computational Modelling of Blood Flow
Hannah Clapham (Singapore)
Within to between host modelling: when, why and how?
In the talk I will discuss the use of linking within to between host modelling for modelling infectiousness and the development and maintenance of immunity to infectious disease.
MS05 Multiscale Modelling of Infectious Diseases
Richard Clayton (Sheffield)
Embedding uncertainty in models of electrical activity in the heart
The heart is an electromechanical pump, where propagating waves of electrical activation act to initiate and synchronise contraction. Abnormal patterns of electrical activation result in disorders of heart rhythm, which may require treatment. Cardiac models aim to reconstruct the electrical activation and recovery of cardiac cells and tissue. They are typically composed of stiff and nonlinear ODEs, which represent electrical activity at the cell scale, coupled through a system of second order PDEs, which describe the propagation of electrical activity. These types of cardiac model are important research tools, and have potential applications for guiding interventions to treat disorders of heart rhythm in patients.
The complexity of cardiac models presents a challenge to their adoption for clinical use. Models are computationally expensive to solve, and are difficult to calibrate to an individual patient using the noisy and often incomplete data available in the clinical setting. We have begun to address these problems for models of electrical activation in the left atrium of the heart. We have developed a method for interpolation of electrical activation time across the left atrium, which takes into account uncertainties in local measurements as well as uncertainties in the interpolation, to yield a probabilistic activation time everywhere. This method has been extended to also produce an uncertain estimate of conduction velocity, which can then be used for inference of parameters for a simplified cell model.
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Bobby Clement (Nottingham)
Two fluids, one magnet. An Unstable Tale.
The phenomenon of interfacial instability has been of growing interest since the initial work presented by Lord Rayleigh in his 1882 paper [1] concerning a dense incompressible fluid being supported by a less dense incompressible fluid. The importance of such instabilities are present when understanding multi-scale flows within many areas of research, from the small scale modelling of Inertial Confinement Fusion (ICF) [2] to the very large scale modelling of bubbles rising in galaxy cluster cooling flows [3, 4]. An example of such an instability is when milk is poured into black tea. We can see an interfacial boundary between the hot water and milk. The boundary is a good example of a density-stratified medium exhibiting a growing instability. Here we follow earlier work [5] by experimentally and theoretically investigating a circular interface between two concentric fluid layers of differing density in a circular domain. The objective is to investigate how instabilities on the interface grow when they are driven by a centrifugal, rather than gravitational, force. Hence, the whole system is rotated about its axis of symmetry. The geometry of the flow domain impose significant technical difficulties in the initial experimental setup of the fluid layers. The fluids are separated uniformly using a strong, externally applied, magnetic field. We will look at the experimental technique used to complete the experiments and then look at some of the data acquired using verified image processing techniques. This will then be compared with a high viscosity Stokes flow approximation using Darcy’s law for a depth averaged velocity in a circular Hele-Shaw cell. Our aim is to provide a suitable first order theoretical prediction of how the interface behaves in an axisymmetric domain.
CT03 Viscous Fluid Dyamis 2
Cecilia Clementi (Freie Universität Berlin)
Designing molecular models by machine learning and experimental data
The last years have seen an immense increase in high-throughput and high-resolution technologies for experimental observation as well as high-performance techniques to simulate molecular systems at a microscopic level, resulting in vast and ever-increasing amounts of high-dimensional data. However, experiments provide only a partial view of macromolecular processes and are limited in their temporal and spatial resolution. On the other hand, atomistic simulations are still not able to sample the conformation space of large complexes, thus leaving significant gaps in our ability to study molecular processes at a biologically relevant scale. We present our efforts to bridge these gaps, by exploiting the available data and using state-of-the-art machine-learning methods to design optimal coarse models for complex macromolecular systems. We show that it is possible to define simplified molecular models to reproduce the essential information contained both in microscopic simulation and experimental measurements.
MS09 Integrating dynamical systems with data driven methods
Matthew Colbrook (Cambridge)
Diagonalising the infinite: How to compute spectra with error control
Spectral theory is ubiquitously used throughout the sciences to solve complex problems. This is done by studying ‘linear operators’, a type of mapping that pervades mathematical analysis and models/captures many physical processes. Just as a sound signal can be broken down into a set of simple frequencies, an infinite-dimensional operator can be decomposed (or “diagonalised”) into simple constituent parts via its spectrum (the generalisation of eigenvalues). Often spectra can only be analysed computationally, and computing spectra is one of the most investigated areas of applied mathematics over the last half-century. Wide-ranging applications include condensed-matter physics, quantum mechanics and chemistry, fluid stability, optics, statistical mechanics, etc. However, the problem is notoriously difficult. Difficulties include spectral pollution (false eigenvalues of finite-dimensional approximations/truncations that “pollute” the true spectrum) and spectral inexactness (parts of the spectrum may fail to be approximated). While there are algorithms that in certain exceptional cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations in applications. It has been an open problem since the 1950s to decide whether such reliable methods exist at all. We affirmatively resolve this question, and we prove that the algorithms provided are optimal, realising the boundary of what computers can achieve. Moreover, the algorithms are easy to implement and parallelise, offer fundamental speed-ups, and allow problems to be tackled that were previously out of reach, regardless of computing power. The method is applied to difficult physical problems such as the spectra of quasicrystals (aperiodic crystals with exotic physical properties).
MS29 IMA Lighthill Thwaites
Heather Collis (Nottingham)
Modelling local auxin biosynthesis in the Arabidopsis root tip
It is estimated that in 2017 nearly 821 million people were living without a secure food supply. Climate change is a common cause of food insecurity that is likely to get significantly worse, but, mathematical modelling can accelerate the development of improved crop varieties that are more resilient to drought and better at acquiring nutrients. Determining how a plant’s root structure develops is crucial in understanding how different varieties and environmental conditions affect a plant’s ability to acquire the nutrients it requires to grow.
The plant hormone auxin plays an important role in many aspects of plant development. Auxin dynamics in the root tip are key to regulating growth, initiating root branching, and producing responses to environmental cues. Using a vertex-based approach,we simulate a system of ordinary differential equations to model auxin biosynthesis, degradation, and transport in real multicellular root-tip geometries. The model incorporates passive and active auxin transport across cell membranes (accounting for experimentally derived distributions of influx and efflux carriers), auxin diffusion between cell cytoplasms via plasmodesmata and auxin diffusion through the cell wall. We use our model to investigate experimental results that show that the root tip of Arabidopsis maintains a significant supply of auxin even when there is no auxin supplied from the shoot.
We optimise the auxin-biosynthesis parameters in our model in order to replicate reciprocal grafting experimental results. We find that a shift from shoot-dominant auxin production towards root-dominant auxin production is required to capture the effect of these experiments. This contradicts current understanding that auxin production in the shoot is dominant and that the root receives most of its auxin via transport from the shoot. We conclude that local auxin biosynthesis in the Arabidopsis root tip plays a critical role in determining auxin levels in the primary root tip.
MS20 Mathematics of the water, energy and food security nexus
Justin Coon (Oxford)
Compressing Spatial Graph Ensembles
Many real-world networks exhibit geometric properties. Brain networks, social networks, and wireless communication networks are a few examples. The storage and transmission of the topologies and structures that define these networks are important tasks, which, given their scale, is often nontrivial. Although some (but not much) information theoretic work has been done to characterize compression limits and algorithms for nonspatial graphs, little is known for the spatial case. In this talk, we will develop a simple information theoretic approach to studying compression limits for a fairly broad class of spatial (random geometric) graphs.
MS08 Spatial Networks
Laura Cope (Cambridge)
Pattern formation in simple jet stream models
Zonal jets are strong and persistent east-west flows that arise spontaneously in planetary atmospheres and oceans. They are ubiquitous, with key examples including mid-latitude jets in the troposphere, multiple jets in the Antarctic Circumpolar Current and flows on gaseous giant planets such as Jupiter and Saturn. Turbulent flows on a beta-plane lead to the spontaneous formation and equilibration of persistent zonal jets. However, the equilibrated jets are not steady and the nature of the time variability in the equilibrated phase is of interest both because of its relevance to the behaviour of naturally occurring jet systems and for the insights it provides into the dynamical mechanisms operating in these systems.
Variability is studied within a barotropic beta-plane model, damped by linear friction, in which stochastic forcing generates a kind of turbulence that in more complicated systems would be generated by internal dynamical instabilities such as baroclinic instability. This nonlinear (NL) system is used to investigate the variability of zonal jets across a broad range of parameters. Comparisons are made with two reduced systems, both of which have received attention in recent years. A quasilinear (QL) model, in which eddy-eddy interactions are neglected, permitting only nonlocal interactions between eddies and the zonal mean flow, is studied in addition to a generalised quasilinear (GQL) system in which certain eddy-eddy interactions are retained. Each system reveals a rich variety of jet variability. In particular, the NL and GQL models are found to admit the formation of systematically migrating jets, a phenomenon that is observed to be robust in subsets of parameter space. Jets migrate north or south with equal probability, occasionally changing their direction of migration.
MS16 Eddies in Geophysical Fluid Dynamics
Nirvana Coppola (Bristol)
Wild Galois representations of hyperelliptic curves
In this talk we will investigate the Galois action on a certain family of hyperelliptic curves defined over local fields. In particular we will look at curves with potentially good reduction, which acquire good reduction over a wildly ramified "large" extension. We will first clarify what large means and then show how to determine the Galois representation in the case considered.
BMC02 Number Theory
Joseph Cousins (Strathclyde)
A static thin ridge of nematic liquid crystal
Industrial applications of nematic liquid crystals continue to evolve, with new emerging technologies such as liquid crystal lenses, whilst older applications such as electronic displays still remain relevant in modern society. The manufacture of these applications often requires the deposition of a nematic droplet on a solid substrate. In these systems there exists a solid-nematic interface between a solid substrate and the nematic, a nematic-isotropic interface (i.e. a free surface) between the nematic and the surrounding atmosphere, and a three-phase contact line. The static continuum description of these interfaces was obtained by Jenkins and Barratt [1] who derived equations for the balance-of-couple and balance-of-stress on the free surface and the force on the contact line in terms of the nematic director and free surface height.
Using the equations defined in [1] we formulate the equations for a thin two-dimensional ridge of nematic in terms of the free surface height h(x) and the director angle θ. Numerical solutions of these thin ridge equations reveal the key role played by the critical Jenkins-Barratt-Barbero-Barbri height hc which has previously been recognised in the context of a uniform layer of nematic contained within solid boundaries [1,2]. We construct solution parameter planes in terms of the anchoring strengths on the substrate and free surface for two scenarios; a pinned ridge and a ridge with contact line described by the nematic Young equation. These solution parameter planes enhance the understanding of static nematic droplets and their potential applications.
[1] J. T. Jenkins and P. J. Barratt, Interfacial effects in the static theory of nematic liquid crystals, Q. J. Mech. Appl. Math. (1974) 111-127.
[2] G. Barbero and R. Barberi, Critical thickness of a hybrid aligned nematic liquid crystal cell, J. de Physique 44 (1983) 609–616.
MS22 Theory and modelling of liquid crystalline fluids
Simon Cox (Aberystwyth)
Bubble entrainment by a sphere falling through a horizontal soap foam
When a solid particle impinges on a soap film it first deforms the film into a catenoid-like shape. As the particle moves through the film, a point is reached at which the film becomes unstable, in a manner familiar to anyone who has formed a soap film catenoid between two rings and pulled them apart. However, in this case the film may not rupture. Instead, the film reforms ("heals") and a small bubble attaches to the particle as it is ejected. This process is fundamental in the use of foams to suppress explosions. We model the quasi-static motion of a spherical particle through a stable horizontal soap film and show how the contact angle at which the soap film meets the particle, as well as the size of the particle itself, influence the size of the bubble that is created. This gives a measure of the energy dissipated by the soap film.
MS24 The mathematics of gas-liquid foams
Elaine Crooks (Swansea)
Invasion speeds in a competition-diffusion model with mutation
We consider a reaction-diffusion system modelling the growth, dispersal and mutation of two phenotypes. This model was proposed in by Elliott and Cornell (2012), who presented evidence that for a class of dispersal and growth coefficients and a small mutation rate, the two phenotypes spread into the unstable extinction state at a single speed that is faster than either phenotype would spread in the absence of mutation. Using the fact that, under reasonable conditions on the mutation and competition parameters, the spreading speed of the two phenotypes is indeed determined by the linearization about the extinction state, we prove that the spreading speed is a non-increasing function of the mutation rate (implying that greater mixing between phenotypes leads to slower propagation), determine the ratio at which the phenotypes occur in the leading edge in the limit of vanishing mutation, and discuss the effect of trade-offs between dispersal and growth on the spreading speed of the phenotypes. This talk is based on joint work with Luca Börger and Aled Morris (Swansea).
MS13 Mathematical challenges in spatial ecology
Konrad Dabrowski (Durham)
Well-Quasi-Orderability on Graphs
A class of graphs is well-quasi-ordered with respect to a containment relation if it contains no infinite set of incomparable elements (an anti-chain) and no infinite decreasing sequence. Being well-quasi-ordered is a highly desirable property, which has frequently been discovered in assorted areas of combinatorics and theoretical computer science. As an example, the Robertson-Seymour Theorem states that the class of all finite graphs is well-quasi-ordered with respect to the minor relation.
We can also consider the question of well-quasi-orderability of graphs with respect to other containment relations. If we consider the induced subgraph relation, the class of cycles forms an infinite anti-chain, so the class of all graphs is not well-quasi-ordered. It is therefore natural to ask what classes of graphs have the property of being well-quasi-ordered with respect to this relation.
This talk will describe some of the recent results in this area and include an introduction to the proof techniques that are available. No prior knowledge of graph classes or well-quasi-orders is needed to understand this talk.
Joint work with Vadim Lozin and Daniël Paulusma.
BMC06 Combinatorics
Mohit Dalwadi (Oxford)
Levitation of a cylinder by a thin viscous fluid
We demonstrate that it is possible to levitate a circular cylinder placed horizontally on a vertical belt covered in a thin layer of oil by moving the belt upwards at a specific speed. The cylinder rotates and is balanced at a fixed location on the belt. Levitation occurs solely through viscous lubrication effects. We present the results of an experimental, asymptotic, and numerical study of this fluid-structure interaction. In particular, we show that understanding the asymptotic structure is, somewhat surprisingly, integral to understanding levitation.
CT08 Industrial fluids
Hannah-May D'Ambrosio (Strathclyde)
Deposition from an evaporating sessile droplet
Hannah-May D’Ambrosio, Stephen K. Wilson, Brian R. Duffy and Alexander W. Wray
The evaporation of sessile droplets occurs in a wide variety of physical contexts, with numerous applications in nature, industry and biology. The key interest in most scientific and industrial processes is the deposit that is left behind after evaporation. For applications such as inkjet printing, the ability to control the distribution of solute during the drying process is extremely important with a particular desire for uniform deposits. Over the past 20 years there has been an explosion of research into the deposition of an evaporating droplet, with particular interest in the well-known coffee-ring effect which results in the solute within the droplet forming a ring deposit at the contact line. In this talk I will investigate the effect of spatial variation of the evaporative flux on the final deposition pattern of an evaporating droplet. I propose a one parameter family of quasi-static evaporative fluxes, which include as special cases diffusion-limited evaporation and uniform evaporation, as well as fluxes with maxima at the centre of the droplet. I solve analytically the general problem for the flow velocity, the concentration of solute inside the droplet and the evolution of the contact line deposit ring. I will show that the deposit is directly linked to the evaporative flux profile with the possibility of either coffee-ring deposits, paraboloidal deposits or mountain/bullseye deposits at the centre of the droplet. I will also examine several interesting cases in detail.
CT14 Droplets
Ben Davison (Edinburgh)
Refined invariants of flopping curves
Associated to a flopping curve in a 3-fold X is a finite-dimensional algebra, the contraction algebra of Donovan and Wemyss, that represents the noncommutative deformations of the structure sheaf of that curve in the category of coherent sheaves on X. A conjecture of Brown and Wemyss states (roughly) that all finite-dimensional Jacobi algebras arise this way. As I'll explain, this conjecture would imply the positivity of all BPS invariants for finite-dimensional Jacobi algebras - these are invariants defined in terms of noncommutative Donaldson-Thomas theory. This is a surprising claim since it is easy to cook up negative DT invariants for infinite-dimensional Jacobi algebras. Finally, I'll explain that this positivity does indeed hold, providing evidence for the conjecture.
BMC07 Algebraic Geometry
Jonathan Dawes (Bath)
Are the Sustainable Development Goals self-consistent and mutually achievable?
The UN’s Sustainable Development Goals (SDGs), launched in 2015, present a global 'to do list' of Challenges. Everyone agrees that there are linkages between them. No two reports agree on what they are. This talk discusses how one might analyse those interlinkages mathematically, and what conclusions we might be able to draw.
Reference:
J.H.P. Dawes, Are the Sustainable Development Goals self-consistent and mutually achievable? Sustainable Development. 2019;1–17. https://doi.org/10.1002/sd.1975
CT 15 Statistical and Numerical Methods
Matt Daws (UCLAN)
An introduction to (quantum) symmetries of (quantum) graphs
We will give a survey about quantum automorphism groups, concentrating upon automorphisms of graphs. We plan to quickly introduce the notion of a compact quantum group, describe how quantum groups can (co)act on spaces and algebras, and then describe a universal construction due to Wang which leads to the idea of quantum automorphisms of a finite set. Viewing a finite (simple) graph as a finite set of vertices with a relation describing the edges allowed Banica to define the compact quantum group of automorphisms of a graph. Surprisingly, this construction has recently appeared, repeatedly, in quantum information theory, and we will give a brief indication of how this is. Time allowing, we will also discuss "quantum graphs", a non-commutative generalisation of a graph, and their quantum automorphisms. The talk will concentrate upon setting the scene and describing some of the technical machinery which occurs.
BMC04 Operator Algebras
Marcelo Goncalves De Martino (Oxford)
Deformation of Howe dualities
I would report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe duality (O(k),spo(2|2)). This Howe dual pair acts on the space of polynomial differential forms on an n-dimensional complex vector space. The Lie superalgebra spo(2|2) considered here contains some important operators such as the Laplacian and the exterior derivative. It is closely related to the Howe dual pair (O(k),sl(2)), which decomposes the space of complex polynomials in n-dimensions in terms of harmonic polynomials. We consider the deformation of these pairs in terms of suitably defined rational Cherednik algebra (and Clifford algebras) and compute the centraliser algebra of the Lie (super)algebras involved. These questions are motivated by the general theory of Dirac operators for Drinfeld algebras laid down by D. Ciubotaru in 2016.
Poster
Alistair Delboyer (Nottingham)
Mathematical Modelling of Heat Pumps As a Renewable Energy Source
Investment in renewable energy has grown significantly to cut carbon emissions and combat climate change. A method to reduce emissions is to utilise heat in the environment to warm or cool buildings using heat pumps. Open water source heat pumps take water from rivers, pass it through the heat pump system, and then discharge it back into the river at a different temperature. The behaviour of arrays of multiple plumes of warm water in strong cross-flows is not well understood, yet may have ecological and efficiency implications. It is important to know how far downstream the effect of these arrays of plumes may travel before becoming negligible in order to inform heat pump placement.
Mathematical models were formulated to study the behaviour of fluid discharged from heat pumps. Initially, the model focused on the behaviour of arrays of plumes in still-water, validated by corresponding experimental data. The model was extended to study thermal plumes in cross-flow, validated by laboratory experiments in a flume that generated turbulent flow similar to that expected in the environment. The models were in close agreement with existing work and experimental observations.
The results are used to predict where there is likely to be a measurable impact of thermal plumes on ambient river temperature, informing the placement of heat pumps and maximising the potential reduction in carbon usage whilst limiting negative impacts on ecosystems.
MS20 Mathematics of the water, energy and food security nexus
Paul Dellar (Oxford)
A discrete Hamiltonian scheme for the multi-layer shallow water equations with complete Coriolis force
Shallow water equations are widely used as conceptual models for studying wave-vortex interactions and other phenomena in atmosphere-ocean fluid dynamics. They almost invariably employ a simplification, the "traditional approximation", that neglects the Coriolis force due to the locally horizontal part of the Earth's rotation vector. We present extended shallow water equations that include the complete Coriolis force. The non-rotating, traditional, and our extended non-traditional shallow water equations can all be formulated as non-canonical Hamiltonian systems that share the same Hamiltonian structure and Poisson bracket, provided one distinguishes between the particle velocity and the canonical momentum per unit mass. The Hamiltonian structure implies conservation laws for energy, momentum, and potential vorticity via Noether's theorem. The Arakawa–Lamb finite difference scheme for the traditional shallow water equations exactly conserves discrete approximations to the energy and potential enstrophy. This increases the robustness of the scheme against nonlinear instabilities, and prevents any distortion of the turbulent cascade through spurious sources and sinks of energy and potential enstrophy. Salmon showed that the Arakawa–Lamb scheme can be re-interpreted as a discrete approximation to the continuous Hamiltonian structure of the traditional shallow water equations. Exploiting the shared Hamiltonian structure, we adapt the Arakawa–Lamb scheme to construct an energy- and potential enstrophy-conserving scheme for our non-traditional shallow water equations. We construct a discrete non-traditional canonical momentum that includes additional coupling between the layer thickness and velocity fields, and modify the discrete kinetic energy to suppress an internal computational instability that otherwise arises for multiple layers. The resulting scheme exhibits the expected second-order convergence under spatial grid refinement. We also confirm that the drifts in the discrete total energy and potential enstrophy due to temporal truncation error may be reduced to machine precision under suitable refinement of the timestep using the third-order Adams–Bashforth and fourth-order Runge–Kutta integration schemes.
This is joint work with Andrew Stewart at UCLA
MS14 Variational Methods in Geophysical Fluid Dynamics
Michael Destrade (Galway)
Growth and remodelling in the mechanics of human brain organoids
Organoids are prototypes of human organs derived from cultured human stem cells. They provide a reliable and accurate experimental model to study the physical mechanisms underlying the early developmental stages of human organs and in particular, the early morphogenesis of the cortex. Here, we propose a mathematical model to elucidate the role played by two mechanisms which have been experimentally proven to be crucial in shaping human brain organoids: the contraction of the inner core of the organoid and the microstructural remodeling of its outer cortex. Our results show that both mechanisms are crucial for the final shape of the organoid and that perturbing those mechanisms can lead to pathological morphologies which are reminiscent of those associated with lissencephaly (smooth brain).
Authors - Valentina Balbi (Galway) , Michel Destrade (Galway) Alain Goriely (Oxford).
MS18 Growth and Remodelling in Soft Tissues
Grisell Diaz Leines (Ruhr)
Comparison of minimum action and steepest descent paths in gradient systems
On high dimensional and complex potential energy landscapes, the identification of the most likely mechanism of a rare reactive transition is a major challenge. The minimum energy path (MEP) is a reaction path (RP) model usually used interchangeably with the steepest descent (SD) trajectory and is commonly associated with the most likely RP. Nevertheless, the local approximation of the MEP that uses only the potential gradient to determine the RP might not always capture all possibilities of curvature and global characteristics of a complex potential energy surface. In particular for energy landscapes with bifurcation points and multiple minima and saddle points the SD path is not unique and can largely differ from the path of maximum likelihood.
Here we compare the SD-based definition of the MEP to the path integral formulation of a trajectory that minimizes an action functional for Brownian dynamics. We show that the minimum action path additionally takes into account the scalar work, which provides a measure of the likelihood of the identified mechanisms in gradient systems. In particular, the scalar work can be used to distinguish between various steepest descent paths in multiple state transitions and to identify the most likely RP near ridges, when bifurcations occur. We show that in systems with non-trivial energy landscapes the evaluation of the action can alleviate some of the limitations of the MEP to model complex transitions, and provides valuable information for the analysis and description of the most likely path.
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Pawel Dlotko (Polish Academy of Sciences)
Computational topology tools in material science
Both topology and material science uses a lot a concept of shape. In this talk I will present computational tools from Topological Data Analysis that can be used in the context of material science.
MS11 Mathematics for Materials Science
Anastasia Doikou (Heriot-Watt)
Set theoretic Yang-Baxter equation, braces and quantum groups
We examine novel links between the theory of braces (nil potent rings) and set theoretical solutions of the Yang-Baxter equation, and fundamental concepts from the theory of quantum integrable systems. More precisely, we make connections with Hecke algebras and we identify quantum groups and admissible Drinfeld twists associated to set-theoretic solutions coming from braces. We also derive new classes of symmetries for the corresponding periodic transfer matrices.
BMC05 Math. Physics
Edward Donlon (TU Dublin)
Dynamics of Generalized Irreversible Sequential Adsorption Models with Applications to Immunosensor Optimization
The biophysical surface phenomenon of Random Sequential Adsorption (RSA) is investigated, during which the irreversible accumulation of surface-active macromolecules, such as proteins, at interfaces occurs. This process underpins the development of many immunoassay technologies, which rely on the physical adsorption of detection antibodies (the adsorbate) on solid substrates (the adsorbent) for binding analytical targets such as pathogenic organisms or biomarkers for cancer diagnostics. Of particular interest for us is the modelling of a generalization of standard RSA termed Cooperative Sequential Adsorption (CSA) where specific chemical reactions between the deposited adsorbates and the adsorbent result in non-uniformly distributed adsorption locations. Starting with one-dimensional analysis, an integro-differential equation that measures the rate of creation and destruction of available deposition positions during the adsorption process is adapted to account for this clustering behaviour. By means of Monte Carlo simulation, the resulting surface area covered (the jamming limit) and the total number of molecules attached to the surface is then studied. Following on from this, the consequences of CSA in two-dimensions is explored.
Poster
Emmanuel Dormy (ENS - CNRS)
Controversial issues concerning the origin of the Earth’s magnetic field
The origin of the Earth’s magnetic field is an apparently simple, yet challenging problem. In terms of mathematical models, dynamo action in a rotating spherical domain is the -now well established- model to account for the magnetic field of planets and stars. Whereas the relevant equations are easily written, the parameters regime relevant to the Earth’s core is so extreme that a numerical solution with these parameters is out of reach of today’s largest computers. This raises the daunting question of the relevance of today’s state-of-the-art numerical models to the mechanisms at work within our planet. Several issues, on which researches often disagree, naturally follow: What can we learn by comparing models with observations? Can we test numerical models against theoretical results? Can the relevant forces balance in the Earth’s core be approached in numerical models? Could the unresolved small scale and rapidly varying flow be important? In this talk, I will try to place the emphasis on open and controversial issues.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
David Dritschel (St Andrews)
Suppression of gravity waves in non-hydrostatic shallow-water turbulence
The Shallow-Water (SW) model is based on two assumptions: (1) the horizontal flow is independent of depth, and (2) the pressure is in hydrostatic balance. The latter, however, need not be imposed, leading to a more accurate model called the Green-Naghdi (GN) model. This model is derived purely by vertically averaging the three-dimensional equations, making only assumption (1). Results are presented comparing the SW and GN models in the evolution of a rotating turbulent flow at moderate Rossby and Froude numbers. We pay particular attention to the generation of gravity waves from the initially balanced vortical flow. Significantly, we find that gravity waves are strongly suppressed in the GN model compared to the SW model. As the only difference between the two models is the inclusion of non-hydrostatic effects, then these effects act to preserve balance. This is unexpected as the gravity waves in the GN model have lower frequencies than in the SW model, and so these waves might be thought to be more easily excited by nonlinear vortical motions.
MS16 Eddies in Geophysical Fluid Dynamics
Moon Duchin (Tufts)
How we divide ourselves up to vote, and why it matters
The UK is the birthplace of the field of political geography, which broadly studies the spatial structures in elections and governance. In the 1970s, an infusion of math jolted the field, bringing new tools for studying the conversion of votes to representation. In the last few years, the mathematicians are at it again, trying to understand the consequences of relying on territorial districts– and making some headway. I'll survey the research frontier, with an emphasis on what's hard but approachable, and explain why some of this work is currently shaking up US politics
Public Lecture (GMJT)
Samantha Durbin (Royal Institution)
Royal Institution Masterclasses
Royal Institution Mathematics, Engineering and Computer Science Masterclasses are hands-on and interactive extracurricular sessions led by top experts from academia and industry for keen and talented young people all around the UK.
The Royal Institution Masterclass programme opens young people’s eyes to the diversity of mathematics, engineering and computer science. Through series of hands-on extra-curricular workshops, students all over the UK meet to explore these subjects in new and exciting ways.
Our Masterclasses are led by inspiring workshop leaders, known as speakers, who come from a variety of STEM (science, technology, engineering and maths) backgrounds. Our speakers share their enthusiasm and experience with the students, nurturing their curiosity. The programme allows students to gain a deeper understanding of the scope, creativity, relevance and potential applications of these far-reaching subjects.
Throughout a Masterclass series, students meet a range of speakers and volunteer helpers, giving them insight into possible careers and helping them to see that the STEM subjects are for everyone. We aim to inspire students to continue their interest and engagement into the future.
We have networks of Mathematics, Engineering and Computer Science Masterclass series at both primary and secondary level. All over the UK, we work with dedicated contributors who take on a variety of roles to make Masterclasses happen in their area.
Further details: www.rigb.org/education/masterclasses/
Outreach Video
Samantha Durbin (Royal Institution)
Talking Maths in Public
Talking Maths in Public is a conference which runs in the UK every two years, for people who work in, or otherwise participate in, communicating mathematics to the public. The event is independently organised, and funded by ticket sales and grants from mathematical institutions. Our events in 2017 and 2019 both fully sold out and were highly rated by attendees.
TMiP is run by an independent committee of people who work in different areas of maths communication, and the event includes workshops provided by expert guests, discussions on varied topics, networking sessions and chances to share ideas and showcase projects.
TMiP 2021 will be held online 25th-28st August 2021, hosted by ICMS Edinburgh
For more details: www.talkingmathsinpublic.uk
Outreach Video
Rob Eastaway (Maths Inspiration)
Maths Inspiration
Maths Inspiration is a national programme of interactive maths lecture shows for teenagers. We give 14-17 year olds a chance to experience the UK's most inspiring maths speakers presenting mathematics live in the context of exciting, real-world applications. Our shows are exceptional value for money.
We usually hold our events in theatres. However, with the restrictions of Covid-19, we will be doing live 'shows' online until theatres re-open. We still hope to be back in theatres in the autumn of 2021 - and maybe sooner.
Maths Inspiration began in 2004. Since then over 150,000 teenagers have attended our shows making us one of the largest and most successful maths enrichment programmes in the UK.
Rob has been Director of Maths Inspiration since it began in 2004. He is an author whose books on everyday maths include the bestselling Why Do Buses Come In Threes? and Maths On the Back of an Envelope. He is the puzzle adviser for New Scientist magazine, and often appears on BBC Radio 4 and 5 Live to talk about the maths of everyday life.
Further information: mathsinspiration.com
Outreach Video
Jonathan Eckhardt (Loughborough)
The peakon resolution conjecture for the conservative Camassa-Holm flow
The Camassa-Holm equation is an integrable nonlinear partial differential equation that models unidirectional wave propagation on shallow water. I will show how it can be proved that global conservative weak solutions form into a train of solitons (peakons) in the long-time limit.
BMC05 Math. Physics
Matthew Edgington (Pirbright Institute)
Population-level multiplexing: A promising strategy to manage resistance against gene drives
A range of gene drive mechanisms have been proposed that are predicted to increase in frequency within a population even when harmful to individuals carrying them. This allows associated desirable genetic material (“cargo genes”) to also increase in frequency. Such systems have garnered much attention for their potential use against a range of globally important issues including vector borne disease, crop pests and invasive species. Perhaps the most high profile mechanisms studied to date are CRISPR-based gene drives which bias inheritance in their favour by “cutting” a target DNA sequence and tricking the system into using a synthetic drive transgene as a repair template - converting drive heterozygotes into homozygotes. Recent studies have shown that alternate repair mechanisms can lead to the formation of cut-resistant alleles and rapidly inducing drive failure. A commonly stated solution is multiplexing - simultaneously targeting multiple sequences at a wild-type locus - since this would require resistance to develop at all target sites for the drive to fail. However, further studies have shown the possibility of simultaneous DNA “cuts” leading to the deletion of large DNA sequences and thus the removal of multiple (if not all) target sequences within a single event. Here we propose and analyse four novel approaches that seek to overcome these issues by multiplexing at the population rather than the individual level. Using stochastic mathematical models we demonstrate significant performance improvements can be obtained using two of these approaches. Based on performance and technical feasibility, we then take forward one preferred approach for further investigation - demonstrating its robustness to a range of performance parameters. We also explore the performance of the system as target population sizes increase, and extrapolate to show that such an approach could potentially allow the targeting of biologically relevant target population sizes.
Poster
Sean Edwards (Manchester)
Localised thermally forced streak solutions for a Falkner-Skan boundary layer
The downstream streak response over a semi-infinte flat plate to upstream localised (weak) thermal forcing is presented. We work in the high Reynolds number Boussinesq regime, which allows for efficient streamwise parabolic marching within the localised boundary region. Steady thermally-forced 3D streaks are embedded in an otherwise 2D Falkner-Skan boundary layer, and we consider the effect of a favourable pressure gradient. It is shown that for sufficiently small upstream temperature forcing, the downstream response is characterised by a bi-global eigenvalue problem. The growth/decay of the streak is shown to be dependent on the Prandtl number and the pressure gradient, and a neutral curve is presented. Finally, the stability of the streak to linear time-harmonic variations in upstream forcing is considered.
CT07 High Reynolds Number
Gennady El (Northumbria)
Dispersive hydrodynamics: old problems and new horizons
Dispersive hydrodynamics---modelled by hyperbolic conservation laws with small dispersive corrections---has recently emerged as a unified mathematical framework for the description of multi-scale nonlinear wave phenomena in dispersive media. The dispersive hydrodynamic framework arises naturally in the description of problems related to the large-scale dynamics of shallow water or internal waves, but also proves to be extremely useful in the modelling of various phenomena in nonlinear optics including the ``atom optics'' of quantum fluids such as Bose-Einstein condensates. The prominent examples of dispersive hydrodynamic phenomena include dispersive shock waves (DSWs), wave-mean field interactions and integrable turbulence. This talk will overview some classical and more recent mathematical and experimental results in this growing field of research. In particular, the effects of non-convexity and randomness will be considered that give rise to plethora of new exciting phenomena.
MS23 Dispersive hydrodynamics and applications
Laila Elatrash (Salford)
Translational small-amplitude vibration of a sphere moving in a uniform flow field at low Reynolds number.
Consider translational small-amplitude vibration of a sphere moving in a uniform flow field at low Reynolds number. This problem is decomposed into a steady solution and a translational vibration solution. The steady solution is given by the near-field Stoke flow past a sphere with a matching far-field Oseen flow. The translational vibration solution is a near-field Stokes flow given by Pozrikidis in terms of an unsteady stokeslet and quadrupole. In this talk, we give the matching far-field Oseen flow by an unsteady oseenlet and quadrupole. The solution is visualized by using matlab, and shown to agree with existing solutions when the frequency of vibration tends to zero (steady solution), and when the far-field length tends to zero (Pozrikidis near-field solution).
CT03 Viscous Fluid Dyamis 2
Meredith Ellis (Oxford)
Predictive models of metabolite concentration for organoid expansion
Organoids are three-dimensional multicellular structures, which are grown in vitro and successfully recapitulate in vivo tissue-specific structure, heterogeneity, and function. Organoid culture is an increasingly relevant technology in biomedical research, with applications in drug discovery and personalised medicine. We are working in collaboration with the biotechnology company Cellesce, who develop bioprocessing systems for the expansion of organoids at scale. Part of their technology includes a bioreactor, which utilises flow of culture media to enhance nutrient delivery to the organoids and facilitate the removal of waste metabolites. A key priority is ensuring uniformity in organoid size and reproducibility; qualities that depends on the bioreactor design and operating conditions. A complete understanding of the system requires knowledge of the spatial and temporal information regarding flow and the resulting oxygen and metabolite concentrations throughout the bioreactor. However, it is impractical to obtain this data empirically, due to the highly-controlled environment of the bioreactor posing difficulties for online real-time monitoring of the system. Thus, we exploit a mathematical modelling approach, to provide spatial as well as temporal information.
In the bioreactor, organoids are seeded as single cells in a thin layer of hydrogel which acts as a porous scaffold. Media is pumped across the top of the hydrogel layer. We wish to identify the optimal flow rate to ensure sufficient delivery of nutrient and removal of waste metabolites. We present a general model for the nutrient and waste metabolite concentrations in the bioreactor. We exploit the slender geometry of the domain and use an asymptotic approach to derive a long-wave approximation of the system. This reduced-order model accounts for the depth-averaged flow, nutrient uptake, and waste metabolite production, and is more analytically tractable.
We explore the behaviour of the long-wave approximation and full system using both analytical and numerical approaches.
Poster
Maclean Jacob Eneotu (Strathclyde)
Fractional flow for reversed flow boundary condition in foam improved oil recovery (IOR)
In foam improved oil recovery (IOR), it is usually the dispersed gaseous phase (that is gas within foam) that displaces the residual oil left behind by natural drive and other primary oil recoverymethods. It is also possible to envisage a reverse case in which the liquid phase (partly oil, but usually containing a significant amount of water plus some surfactant), whose mobility has been shown by several studies to be unaffected by foam injection, is now somehow pushing the invading fluid back; hence flow reversal is occurring. This could happen e.g. if the gas injection pressure declines or alternatively as a new injection well comes online downstream of the foam flow. In that situation, initially foam will be displacing water. Then at a certain time, flow reversal takes place. The simplest model one canwrite for this situation is q_t(t t_r) = −q_t(t t_r. This study is focused on how multiphase (i.e. foam and water) flow in porous media, as described by the fractional flow theory, behaves when this sort of reversal happens. Using the fractional flow model and the method of characteristics (MOC), we have shown that during flow reversal, there is a shock; this is a jump in water saturation `S_w' between foam with a small amount of water (ahead of the shock) and water with a small amount of foam (behind it). The magnitude of the jump in water saturation at the shock grows over time. Depending on how quickly over time the water saturation `S_w' decreases ahead of the shock and how quickly `S_w' increases behind it, the speed of the shock (itself determined by a Rankine-Hugoniot condition or integral mass balance) is found to vary in different ways time. Typically, the tendency is that the shock speed will decrease with time. The position of the shock can also be updated provided the speed is known. Moreover, once an updated position of the shock is specified at any instant in time, so called characteristic fans ahead of and behind it can be used to determine water saturations either side of the shock. Thus it is possible to iterate between determining water saturations across the shock (based on intersections ofcharacteristic lines at the current shock location) and determining (based on those saturations) how fast the shock moves at any given instant and where it will be at a later time. Our study also suggests that during flow reversal in foam IOR, characteristics that start off behind the shock will collide with theshock as they move downstream, whilst the shock itself will collide with characteristics ahead of it. Ultimately, the overall solution to the foam IOR problem will depend on the interaction between the twocharacteristic fans.
This is joint work with Paul Grassia (Strathclyde)
Poster
Jessica Enright (Glasgow)
Networks for modelling in agriculture
Networks are a natural representation of interactions between farms, animals, agricultural managers, or other entities acting in agriculture. I will discuss the application of graph theoretic and network science methods to control disease and pests, and to model the impact of coordinated decision-making on agricultural systems and the landscape in which they are embedded.
MS20 Mathematics of the water, energy and food security nexus
Elaheh Esmaeili (Strathclyde)
Comparison between squeeze film flow of Newtonian and non-Newtonian fluids with applications to foam-formed papermaking process
Foam forming technology for manufacturing paper is a novel technique in which the paper sheets are made from a suspension of pulp fibres in foam, rather than a suspension of pulp fibres in water. As well as reducing the water footprint of the papermaking process overall, foam bubbles along with foam rheology are believed to play a significant role in producing a fibre network with improved characteristics compared to water-formed papers, including more uniform pore size distribution, increased strength, lower density, etc. As a model for foam forming, this work investigates the effect of squeeze film flow of water as a Newtonian fluid versus that of foam as a non-Newtonian fluid between two parallel and non-parallel fibres. The model can help to establish the extent to which foam rheology plays a role in establishing the more uniform pore size distribution of foam-formed papers rather than papers made with water. The hypothesis explored is that the foam might be behaving as a continuum viscoplastic fluid during foam forming, albeit with the viscoplastic fluid properties being related to underlying bubble size. Thus, investigation of squeeze film flow can give an insight into whether non-uniformity of the gap between two fibres being pushed together is reflected in non-uniformity of pore sizes of foam formed paper, and if so, how the non-uniformity of the gap depends on the fluid rheology.
MS24 The mathematics of gas-liquid foams
Elaheh Esmaeili (Strathclyde)
Comparison between squeeze film flow of Newtonian and non-Newtonian fluids with applications to foam-formed papermaking process
Foam forming technology for manufacturing paper is a novel technique in which the paper sheets are made from a suspension of pulp fibres in foam, rather than a suspension of pulp fibres in water. As well as reducing the water footprint of the papermaking process overall, foam bubbles along with foam rheology are believed to play a significant role in producing a fibre network with improved characteristics compared to water-formed papers, including more uniform pore size distribution, increased strength, lower density, etc.
As a model for foam forming, this work investigates the effect of squeeze film flow of water as a Newtonian fluid versus that of foam as a non-Newtonian fluid between two parallel and non-parallel fibres. The model can help to establish the extent to which foam rheology plays a role in establishing the more uniform pore size distribution of foam-formed papers rather than papers made with water. The hypothesis explored is that the foam might be behaving as a continuum viscoplastic fluid during foam forming, albeit with the viscoelastic fluid properties
being related to underlying bubble size. Thus, investigation of squeeze film flow can give an insight into whether non-uniformity of the gap between two fibres being pushed together is reflected in non-uniformity of pore sizes of foam formed paper, and if so, how the non-uniformity of the gap depends on the fluid rheology.
Poster
Jonny Evans (Lancaster)
Contact and symplectic geometry of compound Du Val singularities
Joint work with YankI Lekili. The link of a terminal 3-fold hypersurface singularity (or compound Du Val singularity) is a dynamically convex contact 5-manifold. We compute a contact invariant called the positive symplectic cohomology (SH^+) of the link for a range of cDV singularities including some Brieskorn-Pham examples and for the base of the Laufer flop. Based on this we formulate a conjecture: if the singularity admits a small resolution whose exceptional set has p components, then SH^+ has rank p in every negative degree. Despite the resulting insensitivity of this invariant, we are still able to distinguish many links as contact manifolds by equipping SH^+ with a bigrading.
BMC03 Topology
Alex Evetts (ESI)
Growth and equations in virtually abelian groups
Growth and equations in virtually abelian groups In 1983, Max Benson introduced a framework to study the structure of finitely generated virtually abelian groups, and used it to prove that the growth series of such a group is always a rational function. I will explain this framework (consisting of patterned words and polyhedral sets) and show how it can be extended and modified to study other forms of growth including conjugacy growth, and relative growth. In particular, I will discuss the solution sets to systems of equations in virtually abelian groups (i.e. those group elements which satisfy equations whose constants lie in the group) and demonstrate that their associated relative growth series are rational. Partially based on joint work with Alex Levine.
BMC09 Groups
Maxime Fairon (Glasgow)
Spin Ruijsenaars-Schneider systems from cyclic quivers
The well-known Calogero-Moser system is an integrable system describing the interaction of several particles confined to move on a line. This system admits numerous generalisations, including 'relativistic' analogues called Ruijsenaars-Schneider (RS) systems. In 1995, Krichever and Zabrodin introduced a further extension of the latter systems by endowing the particles with internal degrees of freedom, which are nowadays called spin RS systems.
It is still an open problem to realise how the classical version of all these integrable systems can be obtained geometrically. I will address part of this problem by explaining how to construct the phase space of the (complex) spin trigonometric RS system as a complex manifold associated to a specific family of quivers, which are extensions of a one-loop quiver by one vertex and several arrows. Furthermore, I will outline how replacing this one-loop quiver by a cyclic quiver allows to define new spin generalisations of the trigonometric RS system.
Poster
Enrico Fatighenti (IMT Toulouse)
Fano varieties from homogeneous vector bundles
The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.
BMC07 Algebraic Geometry
Elizabeth Fearon (LSHTM)
Trace and Isolate strategies for the control of SARS CoV-2 in the UK
Test, Trace and Isolate (TTI) interventions are a targeted approach to epidemic control, identifying and isolating suspected or confirmed cases, tracing their exposed contacts and quarantining them to reduce the time in which infected individuals might come into contact with those who are susceptible, leading to onwards transmission. The approach is more targeted than population-wide social distancing measures, but its efficacy for control of SARS-CoV-2 is challenged by pre-symptomatic transmission and the significant proportion of asymptomatic or very mildly symptomatic cases. In the UK, isolating cases and quarantined contacts remain in their homes and there is little effort to prevent within-household transmission. The close and repetitive nature of household contacts increases the risk of infection, but tracing interventions can also take advantage of their structure. Over 2020-2021 we have used a household structured branching process model of infection and contact tracing to investigate how different TTI policies might be made more effective in preventing epidemic growth, key insights of which we summarise here.
MS28 Covid-19 Modelling
Alistair Ferguson (Strathclyde)
The use of Stochastic PDEs in Randomly Layered Elastic Media
This research focuses on modelling linear elastic waves which propagate in a two dimensional random elastic medium. Modelling a propagating wave in random materials is of interest in the application of non-destructive testing where engineers are interested in detecting and measuring cracks and flaws, and in calculating the uncertainty associated with measurements which are obtained by sending a wave into a heterogeneous material. It is not appropriate to apply homogenisation techniques to derive effective medium properties when modelling such applications as the length scales of the probing wave and the scatterers produce a largely incoherent received wave. Instead the analytical framework uses these length scales to derive a diffusion approximation to the elastic wave equation [D. Abrahams, 1992] where a stochastic process models the spatially varying material parameters. The use of limit theorems [Fouque, et al. 2007] for the resulting matrix equations produce stochastic differential equations which have associated infinitesimal generators [B. Øksendal, 2000] which encode information about the stochastic process which couples the wave to the layered microstructure inside the material. The adjoint of this operator feeds into a forward Kolmogorov equation which can be solved to obtain the probability density function for the transmitted and received waves. The moments of these functions can then be used to comment on the uncertainty associated with ultrasonic non-destructive tests in this class of material. The author will present both analytical and numerical results which show the behaviour of a horizontally polarised elastic shear wave, propagating in a two dimensional randomly layered material.
Poster
Sarah Ferguson Briggs (Imperial College)
Linear Stability of a Column of Ferrofluid Centred Along a Rigid Wire
The linear stability of a Newtonian ferrofluid centred on a rigid wire, is investigated in a fully three-dimensional setting. An electric current runs through the wire, generating an azimuthal magnetic field. For both a highly viscous and inviscid system, we consider an inner column of ferrofluid surrounded by another ferrofluid with different magnetic susceptibilities, producing a magnetic stress at the interface of the fluids. When the inner fluid has a larger magnetic susceptibility, the system is linearly unstable to axisymmetric perturbations only, and is stabilised by a sufficiently large magnetic field. When the outer fluid has a larger magnetic susceptibility, we find the system is unstable to both axisymmetric and non-axisymmetric modes. Moreover, in the viscous regime, we find the non-axisymmetric mode to be the most unstable mode. Then, considering a ferrofluid, whose magnetic susceptibility varies radially, we produce stability conditions for axisymmetric and non-axisymmetric modes in both the viscous and inviscid regimes.
CT11 Magnetohydrodynamics
Dmitri Finkelshtein (Swansea)
Structural properties of the mean-field expansion
The classical mean-field scheme is widely used both in statistical physics and in the study of stochastic dynamics of complex systems (individual-based models of population ecology, epidemiology, social sciences etc.). It states that when an appropriately chosen small parameter (e.g. the inverse to the number of interacting elements or a space scale parameter) tends to 0, the second and higher-order spatial correlations factorise within the dynamics, provided that the factorisation took place initially (a.k.a. the propagation of chaos property). Therefore, the spatial correlations, being expanded in the power series w.r.t. the small parameter, have an explicit leading term of the expansion, that is the product of solutions to a certain (nonlinear) kinetic equation. In the talk, I will present a new approach which describes all terms of the expansion in the small parameter through solutions to recurrent systems of linear evolution equations. The approach can be applied to a rather general class of dynamics.
CT 09 Mathematical physics
Barbel Finkenstadt (Warwick)
Spatio-temporal inference on functional variation of mammalian SCN neurons
Circadian rhythms govern physiological functions in most living organisms
and, in mammals, the suprachiasmatic nucleus (SCN) is the main pacemaker of the clock. It consists of approximately 20 K neurons, each acting as noisy molecular clocks that together keep time accurately and without external input. The current understanding of the time-keeping mechanism in the SCN is that of a transcriptional-translational feedback loop where protein products of circadian genes Per and Cry form complexes that inhibit their own transcription. To model bioluminescence-reported gene expression data from organotypic SCN slice cultures, we develop a Bayesian hierarchical model that places a spatial distribution on model parameters. The dynamic model describes deactivation of Cry1 using a recently proposed stochastic model (Calderazzo et al. Bioinformatics 2018) involving a distributed delay that summarises unobserved species in the feedback loop. This constitutes a principled approach to model SCN data where model parameters can be estimated using Markov chain Monte Carlo (MCMC) methodology. We introduce an empirical measure of oscillatory robustness, the posterior probability of a limit cycle, that is consistent with general definitions of biological robustness while incorporating parameter uncertainty. Furthermore, we are able to place the oscillators along a ridge in parameter space that apparently corresponds to a trade-off between resilience to large perturbations and entrainability.
MS04 Stochastic models in biology informed by data
Oliver Fisher (Nottingham)
Multiple target modelling to enable sustainable process manufacturing: An industrial bioprocess case study
Process manufacturing industries constantly strive to make their processes increasingly sustainable from an environmental and economic perspective. A manufacturing system model is a powerful tool to holistically evaluate various manufacturing configurations to determine the most sustainable one. Previously models of process manufacturing systems are typically single target models, trained to fit and/or predict data for a single output variable. However, process manufacturing systems produce a variety of outputs with multiple, sometimes contradictory, sustainability implications. These systems require multiple target models to find the most sustainable manufacturing configuration which considers all outputs.
In this talk I will give an overview of our work modelling a novel bioprocess that treats process wastewaters to reduce pollutant load for reuse, while simultaneously generating energy in the form of biogas was studied. Multiple target models were developed to predict the percentage removal of chemical oxygen demand and total suspended solids, in addition to the biogas (as volume of methane) produced. Predictions from the models were able to reduce wastewater treatment costs by 17.0%. The final model is able to react to new regulations and legislation and/or variations in company, sector, world circumstances to provide the most up to date sustainable manufacturing configuration.
MS20 Mathematics of the water, energy and food security nexus
Nuria Folguera Blasco (Francis Crick Institute)
Role of autoregulatory cascades in fate-decisions
Positive feedback is a common motif in biological systems, as it endows protein and gene networks with interesting features such as signal amplification, modularity and bistability. Although a single positive feedback, given the right parameters, can give rise to these non-linear features, it is often the case that biological systems are decorated with interlinked feedback regulation.
We investigated one particular motif: a chain of auto-regulation, which we deemed Auto-Regulation Cascade (ARC), in the context of cell-fate decisions during early development. In this talk, I will discuss some of the properties (e.g. multistability, noise robustness) and trade-offs of the ARC topology. We propose that chains of positive feedback regulation might be a simple, widespread motif in fate-decisions.
MS04 Stochastic models in biology informed by data
Giada Franz (ETH Zurich)
Free boundary minimal surfaces with connected boundary and arbitrary genus in the unit ball
A free boundary minimal surface (FBMS) in the three-dimensional Euclidean unit ball is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ball (i.e. the unit sphere).
It is natural to ask if there are FBMS in the unit ball of any given genus g and number of boundary components b. Several different examples have already been discovered, but the answer was so far still unknown already in the simple case g=b=1.
In this talk, we will present the construction of a family of FBMS with connected boundary and any given genus. This answers affirmatively to the aforementioned open question.
This is joint work with Alessandro Carlotto and Mario Schulz.
BMC08 Analysis
Hao Gao (Glasgow)
Selection and Parameter Inference of Myocardial Constitutive Laws: from ex vivo to in vivo
Personalized cardiac modelling can provide unique insights on heart function both physiologically and pathologically. Choosing an appropriate constitutive law and inferring its parameters from limited experimental data, however, still remain a great challenge. In this study, we firstly analysed the descriptive and predictive capability of a general 14-parameter constitutive law developed for myocardial passive response using ex vivo data. We then reduced the general laws using Akaike information criterion (AIC) for different experimental studies by maintaining its mechanical integrity whilst achieving minimal computational cost. To model active stress, we introduced a structural tensor into the contraction model for describing dispersed myofibres. Finally we inferred those unknown constitutive parameters using routine clinical data with the constitutive law determined from ex vivo human data. Our results suggest that a combined ex/in vivo experimental and modelling approach is important in selecting an appropriate constitutive law for predictive biomechanical models.
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Jennifer Gaskell (Glasgow)
Approximate multiscale inference for collective animal movement
Often, animal movement data is collected across scales, with some fine-scale information on an individual level from trackers, and population level data from drone images. How do we best infer the underlying movement parameters when given data from both scales? Here, we present a multiscale statistical framework, using Gaussian Process-accelerated Bayesian inference. We investigate the performance when the data collection isn't complete, for example, when only a percentage of the population has been tagged.
MS13 Mathematical challenges in spatial ecology
Aryan Ghobadi (QMU (London))
Skew Braces and Co-quasitriangular Hopf algebras in SupLat
YangBaxter equation, or YBE, first appeared in Statistical Mechanics, and plays a key role in theapplications of Hopf algebras to knot theory and topological quantum field theories. The classificationof YBE solutions on sets was proposed by Drinfeld in the 90s and remains an open problem. Recentadvances in the classification are due to the study of skew braces and their interactions withcombinatorics and ring theory.my recent poster/talk, we will study coquasitriangular Hopf algebras, or CQHAs, in the category SupLatof complete lattices and joinpreserving morphisms. (Based on my recent preprint https://arxiv.org/abs/2009.12815)
Poster
Jeffrey Giansiracusa (Swansea)
Tropical differential equations with non-trivial valuations
About 5 years ago Grigoriev introduced a theory of tropicalizing differential equations and their formal power series solutions over a trivially valued field. In this talk I will describe work with my student Stefano Mereta in which we generalize this theory to allow non-trivially valued fields (such as p-adic fields) and relate the theory to a differential version of Berkovich spaces. Payne proved that the Berkovich analytification is homeomorphic to the inverse limit of tropicalizations, and we prove an analogue of this theorem for differential equations.
MS07 Applied Algebra and Geometry
Francesco Giglio (Glasgow)
Exact equations of state for thermotropic nematics
The Maier-Saupe theory is considered to be a paradigmatic molecular theory for the phase behaviour of uniaxial thermotropic nematic liquid crystals subject to changes of temperature. Several extensions and generalisations of this theory have been proposed since the appearance of the original works by Maier and Saupe in the late 1950s, allowing for the study of a broad variety of liquid crystalline fluids under different physical conditions.
The talk aims at presenting the analysis of a discrete version of the Maier-Saupe model for biaxial nematics subject to changes of temperature and external fields. The statistical thermodynamics of the model for a finite-size system is formulated in terms of an integrable two-component conservation law of hydrodynamic type for the scalar orientational order parameters. In the thermodynamic limit, the system is governed by a hyperbolic conservation law subject to initial conditions specified by the particular molecular geometry. The associated solutions provide the first exact equations of state for nematics subject to external fields, and lead to the derivation of an explicit form of the free energy density in terms of fundamental invariants. Phase transitions are described in terms of the formation and propagation of classical shock waves in the space of thermodynamic variables and their locations identify the phase diagram of the model.
MS22 Theory and modelling of liquid crystalline fluids
Ben Goddard (Edinburgh)
Rigorous results for the overdamped dynamics of complex fluids
Many systems can be modelled as interacting particles suspended in a fluid bath; these are known as complex fluids. Examples include blood, paints, smoke, and jelly. Their dynamics are strongly influenced by: external forces, such as gravity; interparticle forces, such as electrostatics; and hydrodynamic interactions, which are forces generated by the motion of particles, mediated by the bath. We consider the overdamped (high bath friction) limit of such systems, which is relevant in many applications. We first derive a Smoluchowski-type evolution equation for the one-particle distribution function, including a novel definition of the diffusion tensor. After imposing no-flux boundary conditions on a finite domain, we establish conditions that ensure the existence and uniqueness of this nonlocal, nonlinear PDE. We also derive a priori estimates for the rate of convergence to equilibrium, and the stability of these equilibrium states.
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Ben Goddard, Francesca Iezzi, Mary O’Brien (Edinburgh)
Virtual Maths Circles: helping young people to think like mathematicians
The Edinburgh Maths Circles are sessions for children aged 5–16 and their family, run by the University of Edinburgh. The aim is helping children to behave like little mathematicians as they approach and explore open-ended questions (most of which courtesy of nRich: nrich.maths.org/).
In the period 2016–19 Maths Circles were face-to-face drop-in sessions with a free structure. Children were able to tackle whichever problems they liked, at their own pace. University staff and students were at hand to help. Each session would attract an average of 250 visitors.
With the outbreak of Covid 19, we faced the challenge of restructuring the event to make it suitable to run virtually, while remaining faithful to our goals. Now, our Virtual Maths Circles take place every month, with an average of 120 families attending each event, and very positive feedback from participants.
To reach out to people from different backgrounds, we are also running workshops for schoolteachers and community workers, where we share ideas and resources, to empower participants to set up their own Maths Circles. So far more than 400 schoolteachers took part in our workshops, and we have visited many Scottish schools and local libraries.
Watch the video to find out more.
For further information about the University of Edinburgh Maths Outreach program, please visit: www.maths.ed.ac.uk/school-of-mathematics/outreach For further information about Maths Circles please visit: www.maths.ed.ac.uk/school-of-mathematics/outreach/maths-circle
Outreach Video
Magnus Goffeng (Gothenburg)
Exotic examples in Fell algebras
As a partial result towards classifying the C*-algebras that arises from Smale spaces, Robin Deeley and Allan Yashinski studied a Fell algebra arising from Smale spaces with totally disconnected stable sets. The Fell algebra in question has compact spectrum and trivial Dixmier-Duoady class. Using work of Robin Deeley and Karen Strung, the existence of a projection in this Fell algebra was the missing piece needed to finish their classification and a projection was later on (tautologically enough) constructed using classification results. Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant is just a Hausdorff assumption away from being a unital commutative C*-algebra so one might naively think that general approaches from noncommutative geometry will let us put Fell algebras with smooth spectrum on equal footing with manifolds. For instance, one might suspect that all Fell algebras with compact spectrum and trivial Dixmier-Duoady invariant are stably unital. In this talk I will discuss some examples where this fails. Based on joint work with Robin Deeley and Allan Yashinski.
BMC04 Operator Algebras
Irene Gomez Bueno (Málaga)
Collocation methods for high-order well-balanced methods for one-dimensional systems of balance laws
The goal of this work is to develop high-order well-balanced schemes for systems of balance laws. Following two of the authors, we apply a general methodology for developing such numerical methods in which the key ingredient is a well-balanced reconstruction operator (see [1]). A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. This strategy involves a non-linear problem at every cell, at every time step, for every conserved variable, that consists in finding the stationary solution whose average is the given cell value. In the recent paper [2] a fully well-balanced method is presented where the nonlinear problems to be solved in the reconstruction procedure are interpreted as control problems: they consist in finding a solution of an ODE system whose average at the computation interval is given. Our goal now is to present another general implementation of this technique that can be applied to any one-dimensional balance laws based on the application of the collocation RK methods. To check the efficiency of the methods and their well-balancedness, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some nonlinear source terms to the shallow water equations -without and with Manning friction- or Euler equations of gas dynamics with gravity effects. Joint work with M.J. Castro (University of Málaga), C. Parés (University of Málaga), G. Russo (University of Catania).
REFERENCES
[1] M.J. Castro, C. Parés. Well-balanced high-order finite volume methods for systems of balance laws. Journal of Scientific Computing 82, 48, 2020.
[2] I. Gómez-Bueno, M.J. Castro, C. Parés. High-order well-balanced methods for systems of balance laws: a control-based approach. Applied Mathematics and Computation, 394, 125820, 2021.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Francisco Gonzalez Montoya (Bristol)
Phase space structures and escape time for a model for exothermic reactions
In this work, we study a classical Hamiltonian model is a variant of a 2 degree of freedom model proposed to study very exothermic reactions. In particular, the scape time and the phase space structures that control the dynamics for different energies are calculated. The visualization of the phase space sturctures is done with a Lagrangian descriptor based in the classical action.
Poster
Alain Goriely (Oxford)
Morphoelastic rods: Growth and Remodelling in Elastic Filaments
In many growing filamentary structures such as neurons, roots, and stems, the intrinsic shapes and material response are produced by differential growth of the tissue. Therefore, a key problem is to link the growth field at the microscopic level to the macroscopic shape and properties of the filaments. In this talk, starting with a morphoelastic tubular structure and assuming an arbitrary local growth law on the growth tensor, I will give a multiscale method to obtain the overall curvature, torsion, and material parameters of a growing filament. Various examples of curvature and torsion generation are given and the impact of residual stress on the generation of curvature is demonstrated. This theory will then be used to developed a general multiscale theory of tropism in plants.
Authors - A Goriely (Oxofrd), D Moulton (Oxford), H Oliveri (Oxford)
MS18 Growth and Remodelling in Soft Tissues
Tobias Grafke (Warwick)
Metastability in Fluid Turbulence
Fluid in its turbulent state is one of the most complex interacting systems in physics, with an incredibly large number of strongly and nonlinearly interacting degrees of freedom. This becomes different when scale separation introduces coherent, long-lived structures into the fluid flow, which induces order and stability. I will present two very different scenarios of that type: (1) atmospheric jets on gas giants sustained by the turbulent background fluctuations, and (2) turbulent puffs at the onset of turbulence in pipe flows. In both cases, the coherent structure might exist in a multitude of metastable configurations, and turbulent fluctuations drives transitions between them. I will analyze these metastable states, their basins of attraction, separating hypersurfaces, transition states and most likely transition trajectories from the perspective of instanton calculus and Freidlin-Wentzell large deviation theory.
CT07 High Reynolds Number
Raffaele Grande (Cardiff)
Stochastic control problems and evolution by horizontal mean curvature flow
I will briefly introduce the notation of evolution by horizontal mean curvature flow in sub-Riemannian manifolds and in particular in the Heisenberg group, the associated level set equation and suitably associated stochastic control problems. Then I will introduce a regularizing p-problem and show some results on the structure of the optimal controls both for the p-problem and for the $\infty$-problem. At the end I will show some applications of these results to the Riemannian approximation for the horizontal mean curvature flow.
CT05 Optimisation
Paul Grassia (Strathclyde)
Fractional Flow with Flow Reversal: Towards a Model for Flow-Reversed Pressure-Driven Growth
The 2-D propagation of a foam through an oil reservoir is considered during the process of surfactant-alternating-gas improved oil recovery. The model used, the so called 2-D pressure-driven growth model, assumes a region of low mobility, finely-textured foam at the foam front where injected gas meets liquid. The net pressure that is used to drive the foam along is assumed to reduce suddenly at a specific time. Parts of the foam front, deep down near the bottom of the front, must then backtrack: in other words they reverse their flow direction. To describe this process, equations for 1-D fractional flow, the model which underlies 2-D pressure-driven growth, are solved via the method of characteristics. In a diagram of position vs time, the backtracking front is shown to exhibit a complex double fan
structure, with two distinct characteristic fans interacting. One of these characteristic fans is obtained by reflecting a fan that is already present in forward flow mode. The second fan however only appears after the instant of flow reversal. Both fans contribute to the flow's Darcy pressure drop, but the balance of the pressure drop shifts over time from being dominated by the first fan to being dominated by the second. The implications for 2-D pressure-driven growth are that the foam front has even lower mobility in backtracking reverse flow mode than it had in the original forward flow case.
M. Eneotu and P. Grassia. Foam improved oil recovery: Towards a formulation for pressure-driven growth with flow reversal. Proc. Roy. Soc. London Ser. A, 476:20200573, 2020
MS24 The mathematics of gas-liquid foams
James Gregory (Manchester)
A collagen recruitment model of failure in tendons and ligaments
Collagen fibrils are microstructural components of tendons and ligaments that we model as elasto-plastic solids, assumed to be continuously distributed within the tissue. By choosing a simple form for the plastic stress for an individual fibril, we can derive an analytic expression for the stress in the entire tendon and, hence, a macroscopic strain energy function. The model agrees well with experimental data and provides a microscopic description of the macroscopic post-yield behaviour of tendons and ligaments in that the onset of yielding on the macroscale occurs when the first individual fibril yields.
CT 16 Mathematical Biology-3
Ramon Grima (Edinburgh)
Cell size distribution of lineage data: analytic results and parameter inference
Recent advances in single-cell technologies have enabled time-resolved measurements of the cell size over several cell cycles. This data encodes information on how cells correct size aberrations so that they do not grow abnormally large or small. Here we formulate a piecewise deterministic Markov model describing the evolution of the cell size over many generations, for all three cell size homeostasis strategies (timer, sizer, and adder). The model is solved to obtain an analytical expression for the non-Gaussian cell size distribution in a cell lineage; the theory is used to understand how the shape of the distribution is influenced by the parameters controlling the dynamics of the cell cycle and by the choice of cell tracking protocol. The theoretical cell size distribution is found to provide an excellent match to the experimental cell size distribution of E. coli lineage data collected under various growth conditions.
MS04 Stochastic models in biology informed by data
James Grime (Freelance Public Speaker)
How I got started as a public speaker
Mathematician, lecturer, public speaker. A riddle wrapped in a mystery stood next to an Enigma.
In this video, James reveals how he began his career as a public speaker who shares his passion for mathematics in schools, universities, festivals, and through YouTube. Gain valuable insights to turn your mathematical passion into public engagement.
Until early 2020, James could be found touring the world giving public talks. Learn how his projects have evolved in recent times, and thoughts on how they might continue.
Truly essential viewing for anyone with an interest in engaging the public with mathematics.
Further information about James: www.singingbanana.com/
Outreach Video
Michael Grinfeld (Strathclyde)
Explicit solutions and minimality conditions for travelling wave speeds in monostable reaction diffusion equations
In parameterised scalar monostable reaction-diffusion equations it often happens that when pulled fronts exchange minimality with pushed fronts, the pushed front is described by an explicit solution. A priori there is no reason why this should be the case. We formulate sufficient conditions for exact solvability of the travelling wave equations and show, using the Hadeler-Rothe variational principle, that the conditions for the minimality exchange to involve explicit solutions are not generic, giving counterexamples.
Joint work with Elaine Crooks (Swansea).
MS12 Front Propagation in PDE, probability and applications
Mark Gross (Cambridge)
Intrinsic Mirror Symmetry
Mirror symmetry was a phenomenon discovered by physicists around 1989: they observed that certain kinds of six-dimensional geometric objects known as Calabi-Yau manifolds seemed to come in pairs, with a strange relationship between different kinds of geometric objects on the pairs. Since then, the subject has blossomed into a vast field, with many different approaches and philosophies. I will give a brief introduction to the subject, and explain how one of these approaches, developed with Bernd Siebert, has led to a general construction of mirror pairs.
BMC Morning Speaker
Debao Guan (Glasgow)
Effect of Myofibre Architecture on Ventricular Pump Function by Using a Neonatal Porcine Heart Model: from DT-MRI to Rule-based Methods
Myofibre architecture is one of the essential components when constructing personalized cardiac models. In this study, we develop a neonatal porcine bi-ventricle model with three different myofibre architectures for the left ventricle (LV). The most realistic one is derived from ex vivo diffusion tensor magnetic resonance imaging (DT-MRI), and other two simplifications are based on rule-based methods: one is regionally dependent by dividing the LV into 17 segments, each with different myofibre angles, and the other is more simplified by assigning a set of myofibre angles across the whole ventricle. Results from different myofibre architectures are compared in terms of cardiac pump function. We show that the model with the most realistic myofibre architecture can produce larger cardiac output, higher ejection fraction and larger apical twist compared to those of the rule-based models under the same pre/after-loads. Our results also reveal that when the cross-fibre contraction is included, the active stress seems to play a dual role: its sheet-normal component enhances the ventricular contraction while its sheet component does the opposite. We further show that by including non-symmetric fibre dispersion using a general structural tensor, even the most simplified rule-based myofibre model can achieve similar pump function as the most realistic one, and cross-fibre contraction components can be determined from this non-symmetric dispersion approach. Thus, our study highlights the importance of including myofibre dispersion in cardiac modelling if rule-based methods are used, especially in personalized models.
Poster
Adrian Josue Guel Cortez (Coventry)
Information Length Analysis of Linear Autonomous Stochastic Processes
When studying the behaviour of complex dynamical systems, a statistical formulation can provide useful insights. In particular, information geometry is a promising tool for this purpose. In this talk, we introduce the information length for n-dimensional linear autonomous stochastic processes, providing a basic theoretical framework that can be applied to a large set of problems in engineering and physics. Besides, we discuss the application of the metric to a harmonically bound particle system with the natural oscillation frequency Ω, subject to a damping γ and a Gaussian white-noise, exploring how the information length depends on Ω and γ. This elucidates the role of the critical damping γ = 2 Ω in information geometry. Finally, in the long time limit, we show that the information length reflects the linear geometry associated with the Gaussian statistics in a linear stochastic process.
CT 09 Mathematical physics
Carsten Gundlach (Southampton)
Naked singularity formation at the threshold of gravitational collapse
Can naked singularities form in general relativity during the time evolution of regular initial data? It seems the answer is "yes, for generic initial data right on the boundary between collapse (formation of a black hole) and non-collapse". This is well-understood for matter coupled to gravity in spherical symmetry, but much work remains to be done beyond spherical symmetry, and in particular for pure gravity.
CT 09 Mathematical physics
Radhika Gupta (Temple)
Non-uniquely ergodic arational trees in the boundary of Outer space
The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arational trees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will first talk about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.
BMC09 Groups
Parag Gupta (Glasgow)
Effects of Shell Thickness on Cross-Helicity Generation in Convection-Driven Spherical Dynamos
Rotating thermal convection is ubiquitous within the interiors and the atmospheres of celestial bodies. These fluid regions contain plasmas or metallic components so vigorous convection drives large-scale electric currents and produces the self-sustaining magnetic fields characteristic of these celestial objects. Here, we are interested in understanding the relative significance of two key mechanisms for the generation and amplification of magnetic fields, namely the helicity and cross-helicity effects of mean-field dynamo theory. This study aimed to test the hypothesis that the turbulent helicity effects (α-effect) are more significant in the case of geodynamics, while the cross-helicity effect (γ-effect) is more important in the case of the global solar dynamo, due to differences between the shell aspect ratio of the solar convection zone and that of Earth’s inner core. To assess α- and γ- electromotive effects, we performed, an extensive suite of over 40 direct numerical simulations of self-sustained dynamo action driven by thermal convection in rotating spherical fluid shells, where the shell thickness aspect ratio η is varied at fixed values of the other parameters. The simulations are based on the Boussinesq approximation of the governing nonlinear magnetohydrodynamic equations with stress-free velocity boundary conditions. Two distinct branches of dynamo solutions are found to coexist in direct numerical simulations for shell aspect ratios between 0.25 and 0.6 – a mean-field dipolar regime and a fluctuating dipolar regime. The properties characterizing the coexisting dynamo attractors are compared and contrasted, including differences in temporal behavior and spatial structures of both the magnetic field and rotating thermal convection. The helicity α-effect and the cross-helicity γ-effect are found to be comparable in intensity within the fluctuating dipolar dynamo regime, where their ratio does not vary significantly with the shell thickness. In contrast, within the mean-field dipolar dynamo regime the helicity α-effect dominates by approximately two orders of magnitude and becomes stronger with decreasing shell thickness.
Poster
Rod Halburd (UCL)
Finding exact special solutions to non-integrable equations
We will describe methods to find all meromorphic solutions to different classes of (generally non-integrable) equations and how these methods can be extended to other types of functional equations. The analysis here is much more subtle than Kowalevskaya-Painlevé analysis as global methods are needed to track the value distribution of individual solutions. We will discuss how these methods can be modified to allow for branching at fixed singularities.
BMC05 Math. Physics
Cameron Hall (Bristol)
Node-based approximations for contagion dynamics on networks
Contagion models on networks can be used to describe the spread of information, rumours, opinions, and diseases through a population over time. In the simplest contagion models, each node represents an individual that can be in one of a number of states (e.g. Susceptible, Infected, or Recovered), and the states of the nodes evolve according to specified rules. Even with simple Markovian models of transmission and recovery, it is challenging to compute the dynamics of contagion on large networks and approximate models can be very valuable. One of the simplest approximate models is the node-based mean-field approximation or “first-order” approximation. This approximation is obtained by assuming that each node state is independent of the neighbouring node states, leading to a system of ODEs for the node state probabilities. Extensions of the node-based mean-field approximation, such as the pair approximation and other motif-based approximations, can be used to obtain more accurate estimates of contagion dynamics, although at the expense of larger systems and increased complexity. In this talk, I will introduce a modification of the node-based mean-field approximation that can, in certain circumstances, achieve a high degree of accuracy without the need for the larger systems and increased complexity associated with pair-based approximations. This “hybrid” node-based approximation is obtained by combining the classic first-order approximation with an alternative node-based approximation that is exact on trees with a single source of infection. The hybrid approximation exploits the fact that the node-based mean-field approximation always overestimates the speed of infection while the tree-based approximation always underestimates the speed of infection, leading to an approximation that frequently gives a good match to the true dynamics.
MS05 Multiscale Modelling of Infectious Diseases
Madeleine Hall (Imperial College)
Optimal turning gaits for undulators
An organism’s ability to efficiently traverse and search their surroundings can be important to its survival. This has inspired the study of optimal gaits and locomotion strategies, in particular for the case of undulatory movement of slender bodies. The primary focus has been on finding optimal waveforms for moving forwards along straight paths. However, the ability to turn and manoeuvre is also relevant to survival. We revisit this problem in the context of low Reynolds number hydrodynamics and obtain the optimal waveforms for undulators along curved trajectories. For shallow turning angles, we obtain small perturbations of a travelling wave as optimal. For larger turning angles, however, the optimal gait can be radically different, with the undulator abruptly curling and uncurling itself. We believe that these results can lend insight into the search behaviours of simple organisms, such as C. elegans, as well as be a tool for phenotyping their behaviour across mutant strains and under different environmental conditions.
CT01 Viscous Fluid Dyamis 1
Martin Hallnäs (Gothenburg)
Soliton scattering in the hyperbolic relativistic Calogero-Moser system
Integrable N-particle systems of relativistic Calogero-Moser type were first introduced by Ruijsenaars and Schneider (1986) in the classical- and Ruijsenaars (1987) in the quantum case. In the hyperbolic regime they are closely related to several soliton equations, in particular the sine-Gordon equation.
In this talk, I will focus on the quantum case and discuss a proof of the long-standing conjecture that the particles in the relativistic Calogero-Moser system of hyperbolic type exhibit soliton scattering, i.e. conservation of momenta and factorization of scattering amplitudes.
The talk is based on joint work with Simon Ruijsenaars.
BMC05 Math. Physics
Aileen Hamilton (STEM Ambassadors in Scotland)
STEM Ambassadors Programme
STEM Ambassador Programme: life-changing impact for young people, delivered by STEM professionals in classrooms and communities.
STEM subjects are brought to life by over 30,000 volunteers, available across the UK, all free of charge. Inspiring communicators and relatable role models, they are here to help now, by connecting online. Aspirations raised, careers illuminated and learning supported.
Learn more about the STEM Ambassadors through this video. To become one or support our inspiring programme, visit www.stem.org.uk/stem-ambassadors/local-stem-ambassador-hubs
Outreach Video
Florian Hanisch (Potsdam)
Relative Traces in Obstacle Scattering
In obstacle scattering, one is interested in properties of the Laplacian $\Delta$ on the complement of a compact set $\mathcal{O} \subset \mathbb{R}^d$ (the obstacles) with suitable boundary conditions. It may be compared with the free Laplacian $\Delta_0$, defined on all of $\mathbb{R}^d$. For functions $f$ satisfying restrictive assumptions, it is known that differences $f(\Delta) - f(\Delta_0)$ are trace class operators and traces are given by integrals of the Krein spectral shift function associated with $\mathcal{O}$.
We will discuss a relative version of this result. Assuming that $\mathcal{O}$ has two connected components, we look at the setting where both obstacles are present relative to the situation, where one of them has been removed. The former is described by the operator $\Delta$; let $\Delta_1$ and $\Delta_2$ denote the Laplacians after removal of an obstacle. We show that the operator $f(\Delta) - f(\Delta_1) - f(\Delta_2) + f(\Delta_0)$ is now trace class for a much larger class of functions $f$. This is important for physical applications when the choice $f(x) = \sqrt{x}$ corresponds to relative (Casimir) energy densities.
Joint work with A. Strohmaier and A. Waters.
BMC08 Analysis
Reza Haqshenas (UCL)
OptimUS: an open source general purpose ultrasound simulation platform
Therapeutic ultrasound is a non-invasive medical procedure with great potential to transform the treatment of many medical disorders, including Parkinson’s disease, Alzheimer’s disease, and cancer. Biological effects arise from focusing ultrasonic energy to target tissue in the body, without incisions or ionizing radiation. Successful clinical outcomes hugely depend on the ability to optimise ultrasound fields between the ultrasonic source and target tissue. Considering the large dimensions (relative to the wavelengths involved) and inhomogeneity of the target medium, numerical methods for high-performance computation are crucial for optimal therapeutic ultrasound planning. Ultrasonics Group at the Department of Mechanical Engineering, University College London has a long history of developing computational models and promoting a mechanistic understanding of physical and biological effects of ultrasound. This talk will present the latest developments for solving high-frequency high-contrast Helmholtz transmission problems using a novel fast multiple-domain boundary element formulation which is implemented in a software package named OptimUS. OptimUS features an easy-to-use Python front-end, the capability to solve for weak non-linearities, perform constrained optimisation to focus ultrasound through scatterers (such as bone) and to reduce scattering at boundaries where there is significant contrast in tissue properties. OptimUS also provides a simple framework for modelling ultrasonic sources such as planar, bowl and array transducers.
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Heather A. Harrington (Oxford)
Algebraic Systems Biology
Signalling pathways can be modelled as a biochemical reaction network. When the kinetics follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system.
I will overview approaches to analyse these models with steady-state data using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability. Finally, I will present how topological data analysis can provide additional information to distinguish these models and experimental data from wild-type and mutant molecules. These case studies showcase how computational geometry, topology and dynamics can provide new insights in the biological systems, specifically how changes at the molecular scale (e.g. MEK WT and mutants) result in phenotypic changes (e.g.fruit fly mutations).
BMC Morning Speaker
Jacob Harris (Manchester)
Modelling flow in non-uniform Hele-Shaw cells
It is often useful to model fluid flows using simpler expressions than the Navier-Stokes equations to reduce the time and computational resources required to find solutions. In channels with large width to height aspect ratios, considering the height averaged fluid velocity gives a simple yet accurate description of the flow. The most well-known approach yields a single second-order partial differential equation, known as Darcy's law, that describes the fluid behaviour in the small space between two infinite parallel plates (a Hele-Shaw cell). Unfortunately, in this model, no slip and no penetration conditions cannot both be imposed on all boundaries. For problems that are strongly dependent on such conditions, those involving obstructions to the channel or side wall interactions, an ad hoc class of problem-specific, fourth-order equations have been employed instead. These equations often vary in their derivation and are rarely usable outside of their niche. We will propose a generalised fourth order alternative equation, derived from first principles that can give highly accurate descriptions of fluid behaviour in Hele-Shaw cells for a far greater range of geometries than the Darcy's law description.
CT03 Viscous Fluid Dyamis 2
James Harris (Oxford)
Combustion modelling relevant for predicting knock
Knock can cause severe damage to spark-ignition engines. It is characterised by a knocking noise which can be heard outside the engine, and high-frequency oscillations of the pressure within the cylinder. This is understood to be the result of a localised explosion in the unburned gas ahead of the spark-initiated flame. The compression of the gas between the deflagration and the cylinder walls is believed to be the cause of this explosion. We present a model for the combustion of a single species in a one-dimensional combustion chamber with closed ends. We model the behaviour using conservation of mass, momentum and energy, and assume the reaction rate has an Arrhenius temperature dependence. To gain insight into the important processes for knock, we take limits of some of the dimensionless parameters and solve the resulting simplified models.
Poster
Jonathan U. Harrison (Warwick)
Hierarchical Bayesian modelling of chromosome segregation allows characterisation of errors in cell division
Cells divide via a self-organising process known as mitosis where a crucial step is the high fidelity separation of duplicated chromosomes to daughter cells. Errors in segregating chromosomes during cell division are a hallmark of cancer and are associated with developmental syndromes. How cell division achieves high fidelity remains an outstanding question, in particular how errors are detected and corrected. Through automated tracking of chromosomes at fine spatio-temporal resolution over long timescales, we can produce detailed quantification of the behaviour of human cells during mitosis. We propose a force-based stochastic differential equation model, dependent on hidden states governed by a Markov process, to describe the oscillations and segregation of chromosomes in mitosis. By fitting this dynamic model to experimental data in a Bayesian framework, we can infer the timing of the metaphase-anaphase transition (chromosome separation) for each duplicated chromosome pair. By extending this to a hierarchical Bayesian framework, we are able to capture rare reversal events during anaphase in the model. Application of this computational modelling pipeline to experimental data allows lagging chromosomes to be characterised and predicted based on a dynamic signature in metaphase.
MS04 Stochastic models in biology informed by data
Will Hart (Oxford)
A compartmental framework for scaling from within- to between-host epidemiological dynamics
Multi-scale epidemic models have the potential to improve infectious disease outbreak forecasts, by providing a mathematical framework for patient-level (within-host) dynamics to be included in population-scale (between-host) predictions. Forecasts can then be informed by infection data that are collected from patients in cohort studies. Previous multi-scale models have involved integro-differential equations (IDEs). However, IDEs are challenging to solve, requiring bespoke numerical methods. Here we develop an alternative modelling framework, which utilises a multi-stage compartmental model in order to transition from within- to between-host epidemic dynamics. Our framework is mathematically equivalent to previous approaches using IDEs, in the large-compartment limit of our method. However, our approach has the advantage that compartmental models, comprising systems of ordinary differential equations (ODEs), can be solved easily using standard numerical routines and software packages. Applying our method to the case study of influenza A in humans, we explore the predictability of outbreaks in the context of limited and/or inaccurate patient-level data. Our results demonstrate how patient data may be included straightforwardly in outbreak forecasts, as well as how cohort studies can be designed for informing these forecasts. The ease with which our approach can be applied permits its use during future outbreaks of a range of infectious diseases.
MS05 Multiscale Modelling of Infectious Diseases
Stewart Haslinger (Liverpool/Imperial College)
Application of geometrical theory of diffraction to scattering by the tips of rough cracks
Stewart Haslinger (Liverpool), Michael Lowe (Imperial), Peter Huthwaite(Imperial), Richard Craster (Imperial), Fan Shi (Hong Kong University of Science and Technology)
The scattering of elastic waves by the tips and/or edges of smooth cracks is well understood. The geometrical theory of diffraction (GTD) was introduced by Keller [1] and is a useful method to compute asymptotic approximations to diffracted fields, for high frequency and/or large distances from a diffracting inclusion, in both two-dimensional and three-dimensional applications [2,3].
In non-destructive evaluation (NDE), modelling techniques are implemented in the technical justification reports for inspection qualification. The application of GTD for straight and flat defects is well established but in many cases, for example in environments with extremes in temperature and pressure, cracks are likely to be rough. A pragmatic approach is to add a safety factor to compensate for the uncertainty but we use stochastic and numerical methods to derive statistical predictions for the expected diffraction amplitudes as roughness is varied. The models presented here are for the two-dimensional case and include validations using high-fidelity finite element software Pogo [4].
References
[1] J.B.Keller, Geometrical theory of diffraction, JOSA (1962), 52,116–130.
[2] J. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company/American Elsevier, 1973.
[3] J. Achenbach and A.K. Gautesen, Geometrical theory of diffraction for three‐D elastodynamics, Journal of the Acoustical Society of America, (1977), 61(2), 413-421.
[4] P. Huthwaite, Accelerated finite element elastodynamic simulations using the GPU, Journal of Computational Physics, (2014), 257, 687–707, www.pogo.software.
CT12 Waves
Mark Haw (Strathclyde)
From clusters to chains and labyrinths: Structure and kinetics in systems of particles with competing interactions
Even simple interactions in systems of particles—colloids, nanoparticles, proteins—can create surprisingly complex structures and kinetics, with implications for a wide range of applications (eg nanoporous materials, foods, coatings, drug delivery, membranes) as well as fundamental scientific insight (eg the physics of biological systems such as cell membranes). Moreover while the equilibrium state is related to the interaction potential through thermodynamics, many useful systems are not at equilibrium, for example, metastable mixtures of oils and water in foods and personal care products: thus the structures created on the system’s “journey” toward equilibrium, often different to those found at equilibrium, and how that journey can be interrupted, sometimes on very long timescales, i.e., the kinetics of structural change, are important factors. These kinetic aspects are even less straightforward to predict from the interaction potential.
Here, using computer simulation, we explore structures and kinetics in two-dimensional systems of colloids with an arguably ‘simplest case’ complex interaction: a combination of short-range attractive and long-range repulsive potentials. The interaction leads to a range of structural outcomes, such as compact clustering, chain ‘labyrinths’, and coexisting clusters and chains. We explore how different structural measures can lead to useful insight, description and categorisation of the possible structures and how they evolve. Cooperative effects mean the attractive potential, despite being very short-ranged compared to the repulsion, can have significant sometimes counter-intuitive effects on large-scale structure. Kinetics of structural change can also be very sensitive to interactions: for example in some regimes small changes in repulsion range and/or particle area fraction can change timescales of structural evolution by many orders of magnitude.
MS11 Mathematics for Materials Science
Andrew Hazel (Manchester)
Fluid-Structure Interaction, Interfaces and Instabilities In A Simplified Model of Pulmonary Airway Reopening
We study an idealised model of pulmonary airway reopening, previously studied experimentally by Ducloué et al (2017). An elasto-rigid Hele-Shaw cell, a uniform channel of rectangular cross-section with an elastic sheet as the upper wall, is filled with viscous oil and collapsed to a prescribed level by adjusting the pressure difference over the elastic sheet. Air is driven into the channel at constant volume flux which results in the development of an air finger that reopens the channel as it propagates.
We derive a depth-averaged fluid-structure interaction model in the frame of reference of the moving finger, which we explore via steady and unsteady numerical simulations using the software library oomph-lib. As we vary the initial collapse of the channel, the resulting depth variations alter the morphology of the propagating finger and promote a variety of instabilities from tip-splitting to small-scale fingering. We find remarkable direct agreement between our model and Ducloué et al's experiments, and the model has a complex solution structure with multiple stable and unstable steady and time periodic fingers. We conjecture that the interaction between the stable and unstable solutions underpins the complex fingering behaviour.
CT03 Viscous Fluid Dyamis 2
Teresa Heiss (IST Austria)
A Topological Fingerprint for Periodic Crystals
As the atoms in periodic crystals are arranged periodically, such a crystal can be modeled by a periodic point set, i.e. by the union of several translates of a lattice. Two periodic point sets are considered equivalent if there is a rigid motion from one to the other. A periodic point set can be represented by a finite cutout s.t. copying this cutout infinitely often in all directions yields the periodic point set. The fact that these cutouts are not unique creates problems when working with them. Therefore, material scientists would like to work with a complete, continuous invariant instead. We conjecture that a tool from topological data analysis, namely the sequence of order k persistence diagrams for all positive integers k, is such a complete, continuous invariant of equivalence classes of periodic point sets.
MS11 Mathematics for Materials Science
Nathaniel Henman (UCL)
Pre-impact dynamics of a droplet impinging on a deformable surface
The non-linear interaction between air and a water droplet just prior to impingement on a surface is a phenomenon that has been researched extensively and occurs in a number of industrial settings. The role that surface deformation plays in an air cushioned impact of a liquid droplet is considered here. Assuming small density and viscosity ratios between the liquid and air, a reduced system of integro-differential equations is derived governing the liquid droplet free-surface shape and the pressure in the thin air film. To close the system, a membrane-type model is used for the shape of the deformable surface. The magnitude and shape of the surface deformation is determined by the surface properties, including surface stiffness. We solve the set of governing equations numerically and present a parametric study by varying the stiffness of the surface. It is found that lowering the surface stiffness results in reduced contact pressure, but bubble entrapment still occurs and in fact increases in magnitude (for a water droplet of radius 1 mm and approach speed of 1 m/s, a reduction in surface stiffness corresponding to a maximum surface deflection of 6.05 µm results in a 29% increase in the initial horizontal bubble extent and a 40% increase in the height, when compared to an impact with a flat rigid surface). Finally, a case is considered in the limit of large surface deformations, where an implied simpler pressure-shape relationship leads to a large time analysis, with good agreement to the numerical results. A quantitative connection with recent experiments is also found.
CT08 Industrial fluids
David Henry (Cork)
Exact, free-surface Equatorial flows
We construct a new exact solution to the governing equations for geophysical fluid dynamics, in the equatorial region of the ocean. The solution is presented in the terms of spherical coordinates, and represents a steady purely-azimuthal flow with a free-surface. Of particular note is that this solution accommodates a general fluid stratification: the density may vary both with depth and with latitude. Using a short-wavelength stability analysis we prove that flows defined by our exact solution are stable for a certain choice of the density distribution. This is joint work with Calin Martin, University of Vienna.
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Sophie Hermann (Bayreuth)
Non-negative interfacial tension in phase-separated active Brownian particles
We present a microscopic theory for the nonequilibrium interfacialtension of the free interface between gas and liquid phases of active Brownian particles. We split the exact force balance equation into flow and structural contributions, and perform a square gradient treatment of the relevant contributions to the internal force field. This approach is general and applies to inhomogeneous nonequilibrium steady states.We find the interfacial tension to be unique and to be positive [1], which opposes claims based on computer simulations [2] and delivers the theoretical justification for the widely observed interfacial stability in active Brownian dynamics many-body simulations. Our treatment is based on the quiet life mechanism [3] for active phase separation and it implements the power functional concept [4,5]. We discuss the relevance of the polarization that occurs at the interface [6] and in general steady states [7].
References:
[1] Non-negative interfacial tension in phase-separated active Brownian particles, S. Hermann, D. de las Heras and M. Schmidt, Phys. Rev. Lett. 128, 268002 (2019).
[2] Negative interfacial tension in phase-separated active Brownian particles, J. BialkÈ, J. T. Siebert, H. Lˆwen, and T. Speck, Phys. Rev. Lett. 115, 098301 (2015).
[3] Phase coexistence of active Brownian particles, S. Hermann, P. Krinninger, D. de las Heras and M. Schmidt, Phys. Rev. E 100, 052604 (2019).
[4] Power functional theory for Brownian dynamics, M. Schmidt and J. M. Brader, J. Chem. Phys. 138, 214101 (2013).
[5] Power functional theory for active Brownian particles: general formulation and power sum rules, P. Krinninger and M. Schmidt, J. Chem. Phys. 150, 074112 (2019).
[6] Exact sum rules for active polarization, S. Hermann and M. Schmidt (to be published).
[7] Active ideal sedimentation: Exact two-dimensional steady states,S. Hermann and M. Schmidt, Soft Matter 14, 1614 (2018).
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Fritz Hiesmayr (University College London)
Asymptotics of two-valued minimal graphs
A two-valued function is a function defined on a subset $\mathbb{R}^n$ and taking values in the set of unordered pairs of real numbers. We consider its graph as a subset of $\mathbb{R}^{n+1}$, with two (possibly equal) points lying above every point in the domain. This is called minimal if it is a critical point for the area functional, and the function satisfies a quasilinear elliptic PDE. Two-valued minimal graphs are of interest as they provide perhaps the simplest non-trivial setting where the challenging issue of branch point singularities arises. We present recent work on entire two-valued minimal graphs, including results concerning their rigidity in low dimensions.
BMC08 Analysis
Des Higham (Edinburgh)
Accuracy and Stability Issues in Deep Learning
The numerical analysis community has a well-developed set of theory and tools for defining, analyzing and experimentally testing algorithms in terms of accuracy and stability. I will discuss how these tools can be applied in the context of training and evaluating artificial neural networks. Such ideas are relevant to the existence of "adversarial examples" that have been shown to fool many image classifiers. I will also discuss the role played by high-dimensionality.
MS02 Mathematics for Data Science
Sascha Hilgenfeldt (Illinois)
Foams: A Prototype for the Structure and Dynamics of Interfaces
Liquid foams are fascinating materials that provide study cases to mathematicians, physicists, engineers, and biologists. The increasing importance of mechanical and dynamical processes involving interfaces on small scales in both hard and soft condensed matter has prompted questions that can often be most rigorously studied in foams, where interfaces are mechanically simple. We will show how modeling tools applied to foams allow for quantitative insight into two very different systems: (i) The mechanical energy landscape of metastable states in cellular matter such as biological tissues is complicated and poorly understood. A new approach infers mechanical energy from statistical information of the disordered structure. Results from foams can be quantitatively translated to other systems with much more complicated energy functionals, to elucidate the mechanics of tissue layers purely from visual snapshot information. (ii) Liquid foams have long been used as model systems for rearrangements in atomic lattices. It is shown here that they can be used to model fracture events as well, providing the first detailed experimental and theoretical insight into the existence of a velocity gap, i.e., a minimum speed required for steady propagation of a crack. Modeling based on fluid dynamics phenomena such as film instability and non-linear dissipation results in highly dynamical, robust fracture behavior that predicts the stress distribution and speed of propagating cracks.
Spanning such a wide range of physical phenomena both static and dynamic, material properties both solid and liquid, and structures both discrete and continuum, illustrates that foams are an enormously versatile and vital tool for interfacial modeling, capable of yielding new modeling approaches as well as insight of practical use.
Plenary (Stewartson Lecture)
Nick Hill (Glasgow)
Discrete-to-continuum modelling of cells to tissue
Living tissues are composed of large numbers of cells packed together within an extracellular matrix. In order to understand the process of growth and remodelling in soft tissues that are subject to internal and external forces and strains, multiscale models that describe the interactions between individual cells and the tissue as a whole are needed. A significant challenge in multiscale modelling of tissues is to produce macroscale continuum models which rationally encode behaviour from the microscale (discrete cells). Over the years a number of highly successful approaches have been developed to rationally form macroscale models for multiscale processes such as solute transport and cell-cell signalling. However, such approaches have focused on homogenization techniques, which typically rely on underlying symmetries or periodicity on the smaller scales. We address the need for models that rationally incorporate the underlying mechanical properties of individual cells, without assuming homogeneity, symmetry or periodicity at the cell level. This challenge is particularly pertinent in modelling cardiac tissue, where the individual cells experience significant mechanical deformation in response to (periodic) electrical signalling. In particular, we are interested in cases where the mechanical properties of the cardiac cells may vary significantly between different regions of the heart (e.g. in disease or following a myocardial infarction).
We consider a single line of nonlinearly hyperelastic cells of finite size, with forces transmitted across the boundaries between neighbours. One or both ends of the line are fixed to represent free expansion or confinement. The dynamics of the array is given by a system of discrete 1D ODE's. Individual cells grow in volume and divide into two identical daughter cells. The parent cell divides its mass equally so that each daughter cell is half the total length of the parent cell, and an extra boundary at the midpoint of the parent cell is introduced. Two examples of resistance to motion are considered.
Firstly, we suppose that the cells are binding and unbinding to a fixed substrate, providing a resistive force is proportional to the speed of the cell relative to the substrate (Stokes dissipation). Secondly, we consider a local resistance to motion arising from the motion of a cell boundary relative to its neighbours so that the damping force is proportional to the rate of elongation of the cell (Kelvin dissipation).
Having constructed and solved the discrete model, we then use the methods of discrete-to-continuum upscaling to derive new PDE models using Taylor expansions local to each discrete cell, which requires that the properties of the individual cells (e.g. shear modulus) vary smoothly along the array. The discrete and PDE models are solved computationally for a range of imposed boundary and growth conditions.
We demonstrate excellent agreement between the solutions (including diagnostics such as pressure and stretch along the array) of the discrete and PDE models for a number of examples, including a ring of cells (e.g. myocytes) with a wave of active contraction, growth of incompressible neo-Hookean cells, and stress-dependent growth. Qualitative differences are found in long-time scaling laws for the growth of the array of cells for stress-dependent and independent cell division rules.
These methods provide a rational multiscale approach for deriving continuum models for soft tissues based on measured properties of individual cells. They can be extended to 2D and 3D.
Authors - Nick Hill (Glasgow) Roxanna Barry (Glasgow), Peter Stewart (Glasgow)
This work was funded by the U.K. EPSRC SofTMech Centre for Mathematical Sciences in Healthcare (EP/N014642/1).
MS18 Growth and Remodelling in Soft Tissues
Daniel Hill (Surrey)
On the existence of localised radial patterns on the surface of a ferrofluid
Ferrofluids, magnetic fluids containing iron nanoparticles, provide a good experimental medium to investigate properties of nonlinear waves and coherent structures. For a vertically applied magnetic field, there exists a critical field strength at which a surface instability occurs; here, spikes emerge from the ferrofluid and arrange in domain-covering cellular patterns. In 2005, solitary spikes were experimentally observed. These spikes were not affected by the shape of the fluid's container and drifted around the domain, indicating that they were localised solutions.
In this talk, I will introduce the ferrohydrostatic model, formulated as a free-surface problem, and present our formal results for showing the existence of localised radial solutions via local invariant-manifold theory. This includes the introduction of an appropriate 'spectral' decomposition to reduce the problem to infinitely-many ODEs and then employing geometric blow-up coordinates to identify exponentially decaying solutions. Finally, I will highlight the three classes of localised radial solutions found in the ferrohydrostatic problem and explore the parameter regions in which these patterns emerge.
MS29 IMA Lighthill Thwaites
Alice Hodson (Warwick)
Construction of projectors for the conforming and non-conforming virtual element method
The virtual element method (VEM) began as an extension of the finite element method (FEM), and has gained a lot of attention since its first appearances in the literature. VEM is desirable to use in place of FEM due to the method allowing the considered domain to be decomposed into more general polygonal meshes. The ease in which VEM spaces can be constructed to handle higher order continuity conditions is another highly desirable property.
In contrast to the finite element setting, it is unclear whether it is possible to create a uniform framework to generalise the virtual element setting. Since each VEM construction depends on the prospective problem, the projections and discrete forms need to be set up each time. We explore the possibilities of defining the projection operators in a way that; (a) does not depend on the bilinear form, and (b) are computable using only the degrees of freedom. This would then ease implementation and create an elegant theory for the VEM framework.
The aim of this talk is to describe the generalisation of the construction of the projectors needed in the virtual element discretisation of polyharmonic problems. The generalisation is considered in two spatial dimensions, in both the conforming and non-conforming cases.
CT 15 Statistical and Numerical Methods
Andrew Hone (Kent)
Heron triangles with two rational medians and Somos-5 sequences
Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. In fact Schubert made the erroneous assertion that even two rational medians was impossible, but Buchholz and Rathbun later showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve $E(\Q)$ with Mordell-Weil group $\Z/2\Z\oplus\Z$, and they observed an apparent connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths and the area in this infinite family of Heron triangles with two rational medians.
BMC05 Math. Physics
Curtis Hooper (Loughborough)
Undular bores generated by fracture
Abstract
We demonstrate for the first time, using high-speed pointwise photoelasticity, the generation of undular bores in solid (polymethylmethacrylate) pre-strained bars by fracture [1]. The development of the oscillations is strongly dependent on the strain rate of the release wave which propagates outwards from the fracture site. Both the nonlinear Gardner equation and its linearised (near the level of the pre-strain) version are used to model the wave propagation, and are shown to provide good agreement with the key observed experimental features for suitable choice of elastic parameters [1]. We also consider viscoelastic corrections which bring the modelling results in an even better agreement with experiments. The experimental and theoretical approaches presented open new avenues and analytical tools for the study and application of dispersive shock waves in solids.
Reference:
[1] C.G. Hooper, P.D. Ruiz, J.M. Huntley, K.R. Khusnutdinova, Undular bores generated by fracture, arXiv: 2003.06697v2 [nlin.PS] 4 Aug 2020.
Authors: C.G. Hooper, P.D. Ruiz, J.M. Huntley, K.R. Khusnutdinova
CT17 Solid Mechanics
Susanne Horn (Coventry)
Tornado-like Vortices in Coriolis-Centrifugal Convection
Buoyancy and rotationally driven flows are ubiquitous in nature and they play an important role in a wide range of geophysical phenomena. Rotating Rayleigh-B\'enard convection, a fluid layer heated from below, cooled from above and rotated around its vertical axis, serves as an idealised model system for the underlying flow physics.
In most studies, rotation has only been considered in terms of the Coriolis force, whereas the centrifugal force has been neglected. Hence, it remains largely unknown how flows are altered by centrifugal buoyancy, in particular, in the turbulent regime.
We have begun to fill this gap by numerically characterising rotating convection including the full inertial term, i.e., by including both Coriolis and centrifugal forces (Horn & Aurnou, Phys. Rev. Lett. 120, 2018). Our work has revealed that in Coriolis-centrifugal convection storm-like structures can develop, ranging from eyes and secondary eyewalls found in hurricanes and typhoons, to concentrated helical upflows characteristic of tornadoes.
Here, we will mainly focus on the tornado-like vortices. These vortices are not only self-consistently generated, but also exhibit the physical and visual features of type I tornadoes, i.e. tornadoes that form within mesocyclones contained in supercell thunderstorms. Using a suite of direct numerical simulations, we demonstrate that Coriolis-centrifugal convection is a suitable idealised model for mesocyclone systems that also captures principal tornado features.
MS16 Eddies in Geophysical Fluid Dynamics
Lewis Hou (Science Ceilidh)
Science Ceilidh
The Science Ceilidh is an award-winning organisation connecting creativity, culture, STEM, research and health and wellbeing with and by community groups, libraries, youth clubs and schools across Scotland.
We are wee, but have a big vision to support a Scotland where everyone is empowered to explore, play, experiment, develop and apply their creativity, skills and curiosity to bring about positive and sustainable change, learning, connectedness, kindness and joy in their communities.
• How are maths and creativity connected? Maths Week Scotland with Science Ceilidh
• How many symmetries are in dances? Maths Week Scotland with Science Ceilidh
Further information: www.scienceceilidh.com/
Outreach Video
Thomas House (Manchester)
Household transmission models of COVID-19 from community data over time in England
The response of many governments to the COVID-19 pandemic has involved measures to control within- and between-household transmission, providing motivation to improve understanding of the absolute and relative risks in these contexts. I will present methods for and results from exploratory, residual-based, and transmission-dynamic household analysis of the Office for National Statistics (ONS) COVID-19 Infection Survey (CIS) data from 26 April 2020 to 8 March 2021 in England
MS28 Covid-19 Modelling
Sam Hughes (Southampton)
Graphs and complexes of lattices
We will survey the theory of lattices in the isometry groups of CAT(0) spaces. We will then introduce the notions of a graph and complex of lattices, these describe a framework for studying a class of CAT(0)lattices containing BurgerMozes groups, LearyMinasyan groups and Sarithmetic lattices. Finally, we will present a number of applications.
Poster
Eugenie Hunsicker (Loughborough)
Image manifolds and data integration
Over the past thirty years, mathematical modelling of materials has produced tremendous insights into the self-organisation of matter at the nano scale into complex structures. This can occur through a variety of mechanisms, including evaporation of a solution to leave behind a pattern of nanoparticles on a surface, called evaporative deposition. If we could harness these natural mechanisms, we could create new structured materials and nano-devices in a cost, time, and energy-efficient manner. Current research has produced numerical models of these processes, which explain how the underlying physics generates the variety of structures observed. As yet, however, there is only a general qualitative understanding of how these models relate to experiments and how structures depend on the conditions under which they are produced. Bridging the current gap between model and experiment will require a rigorous quantitative method for fitting the models to experimental data.
If we want to harness these mechanisms for industrial applications and use the models to predict the conditions required to produce desirable structures: How variable are the structures created under the same or similar experimental conditions? How tightly do conditions need to be controlled in order to repeatably produce structures that are similar enough to be interchangeable for industrial uses? How can we determine the correct laboratory conditions to produce specific structures?
To answer these questions, we need to be able fit the model statistically to the data. Fitting models to data involves minimising the error (i.e., distance) between predicted data and real data. Namely, it requires a meaningful way to calculate distance in the data space. As both the models and the data come in the form of complex structures, this involves computing distances between structures using a mathematical description of the geometry of the structure space, which will in turn relate to the geometry of the structures themselves.
This talk will discuss preliminary work on a method to embed simulated image spaces into high dimensional feature spaces in order to permit the development of meaningful image metrics for data integration.
MS07 Applied Algebra and Geometry
Matthew Hunt (Warwick)
Free surface flows in electrohydrodynamics with prescribed vorticity.
Free surface flows have classically used the assumption of irrotationality in the derivation of various different models which includes the celebrated Korteweg-de Vries(KdV) equation. In recent years there has been interest in the inclusion of constant vorticity to the model. This has lead to the use of various potentials to try and get a model described the free surface.
The approach taken here is to isolate the vertical component of the velocity and use that as the master equation. This will allow the introduction of not only a constant vorticity but a variable one. The results presented today will be a derivation of the method, linear and weakly nonlinear theory for a variety of different vorticity distributions for both linear and weakly nonlinear cases.
CT11 Magnetohydrodynamics
Robert Insall (Beatson Institute)
Self-generated gradients - cells solving mazes and other biological problems by generating their own steering gradients.
Chemotaxis is fundamentally important in biological processes from embryogenesis to immune function. One of the most important questions, never asked as often as it should be, is “where do chemotactic gradients come from?”. We have used an interative process of modelling and wet experiments to show that cells very often participate in making the same gradients that they respond to. We show how this is important in cancer spread, standard chemotaxis assays, and other scenarios where we had thought chemotaxis was simple.
MS06 Tracking cellular processes through the scales
Edward Johnson (UCL)
Wavepackets in the anomalous Ostrovsky equation
The anomalous Ostrovsky equation, which describes waves in vertically sheared ocean flows and magneto-acoustic waves, possesses steadily propagating, finite-amplitude, localised wavepacket solutions. It is shown here that these solutions can be obtained asymptotically, using Whitham modulation theory, as the solution to a nonlinear eigenvalue problem. This allows the various wavepacket solutions to be delineated and compared to solutions of the full equations of motion. A periodic solution with an embedded wavetrain is also constructed.
MS23 Dispersive hydrodynamics and applications
Laura Johnson (St. Andrews)
Partitioning groups into External Difference Familes and other similar structures
An External Difference Family (EDF) is a combinatorial structure formed by disjoint subsets of a group G, such that each element within G occurs lambda times as a difference between elements of disjoint subsets. We consider various questions about EDF's and EDF-like structures, including their relationship with traditional difference families and difference sets. We also look at the implications for the internal differences of each subset, if the disjoint subsets of an EDF-like object partition the whole group G or a proper subset of G.
Poster
Connah Johnson (Warwick)
ChemChaste: A computational tool for multiscale simulations of chemical coupled cell populations
Many biological systems are spatially organised, from animals and plants to microbial communities. The development of computational tools for modelling spatially organised biological systems has largely focused on either so-called individual-based models or on physico-chemical reaction-diffusion models. Individual-based models can incorporate cell specific properties and rules, such as cell cycle dynamics, replication, and speciation. However, these models allow limited resolution for chemical dynamics and related chemical mechanisms. In contrast, reaction-diffusion models allow finer simulations of chemical dynamics using density state variables. However these models often assume basic functional forms for the chemical reactions and lack the biological cell-specific attributes limiting mechanistic insight. Here, we combine the benefits of individual-based and reaction-diffusion models, by extending an existing software library Chaste. Chaste is a modular, open-source PDE solver platform that is already widely used by the systems biology community.
Here, we introduce ChemChaste expanding the Chaste functionality to allow the simulation of complex reaction-diffusion dynamics with multiple state variables and multiple cell structures. We aim to use this extended Chaste platform to simulate early evolution of protocellular metabolic systems, in particular, reaction systems that are separated across cell-like phase separations in an otherwise homogenous primordial soup. Spatial dynamics in such early metabolic systems have not been considered to date and it will be interesting to characterise what kind of system dynamics can emerge under different parameter regimes of metabolite diffusion, phase dynamics, and reaction kinetics.
Poster
Eleanor Johnstone (Manchester)
Free-stream coherent structures in parallel compressible boundary-layer flows at subsonic Mach numbers
As a first step towards the asymptotic description of coherent structures in compressible shear flows, we present a description of nonlinear equilibrium solutions of the Navier--Stokes equations in the compressible asymptotic suction boundary layer (ASBL). The free-stream Mach number is assumed to be less than or equal to 0.8 so that the flow is in the subsonic regime and we assume a perfect gas. We extend the large-Reynolds number free-stream coherent structure theory of Deguchi & Hall (2014) for incompressible ASBL flow to describe a nonlinear interaction in a thin layer situated just below the free-stream, which produces streaky disturbances to both the velocity and temperature fields that can grow exponentially towards the wall. We complete the description of the growth of the velocity and thermal streaks throughout the flow by solving the compressible boundary-region equations numerically. We show that the velocity and thermal streaks obtain their maximum amplitude in the unperturbed boundary layer. Increasing the free-stream Mach number enhances the thermal streaks, whereas varying the Prandtl number changes the location of the maximum amplitude of the thermal streak relative to the velocity streak. Such nonlinear equilibrium states have been implicated in shear transition in incompressible flows; therefore, our results indicate that a similar mechanism may also be present in compressible flows.
MS29 IMA Lighthill Thwaites
Anna Kalogirou (Nottingham)
Instabilities at a sheared interface over a liquid laden with soluble surfactant
TThe linear stability of a semi-infinite fluid undergoing a shearing motion over a fluid layer that is laden with soluble surfactant and that is bounded below by a plane wall is investigated under conditions of Stokes flow. While it is known that this configuration is unstable in the presence of an insoluble surfactant, it is shown via a linear stability analysis that surfactant solubility has a stabilising effect on the flow. As the solubility increases, large wavelength perturbations are stabilised first, leaving open the possibility of mid-wave instability for moderate surfactant solubilities, and the flow is fully stabilised when the solubility exceeds a threshold value. The predictions of the linear stability analysis are supported by an energy budget analysis which is also used to determine the key physical effects responsible for the (de)stabilisation. Asymptotic expansions performed for long-wavelength perturbations turn out to be non-uniform in the insoluble surfactant limit. In keeping with the findings for insoluble surfactant obtained by Pozrikidis & Hill (2011), the presence of the wall is found to be a crucial factor in the instability.
This is joint work with Mark Blyth (University of East Anglia).
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Alex Kartun-Giles (NTU Singapore)
The beautiful geometry of spatial networks
The geometry induced by the length of shortest paths in a network is by no means the only geometry. By studying the renormalisation group, among other methods from statistical physics, self-similar fractal geometry, hyperbolic geometry, and geometry based on communicability and diffusion have been discovered (Song et al, Nature 433, 2005). However, that is not the end of the story. Deep ideas from geometry such as analogues of the Erlangen program of Felix Klein for networks suggest that network geometry has yet to reveal much of its depth. This is particularly true when we consider the various forms of random geometry found in random spatial networks. We discuss this in the context of spatial complex networks, which derive their geometric features from the underlying topology of their embedding space, but yet posses their own unique and beautiful network geometry, manifest in famous examples including the growth of megacities, flocking starlings, stem cells, and the brain.
MS08 Spatial Networks
Jon Keating (Oxford)
Random Matrix Theory, Integrable Systems, and Discrete Probability
I will review some connections between Random Matrix Theory, aspects of Discrete Probability, and certain integrable systems. I will then outline how these connections have led to recent progress in computing various moments of characteristic polynomials of random matrices that had previously been inaccessible.
BMC Morning Speaker
Ailsa Keating (Cambridge)
An invitation to symplectic mapping class groups
Given a symplectic manifold, one can define its symplectic mapping class group: diffeomorphisms preserving the symplectic form, up to symplectic isotopy. In dimension two, this agrees with the classical mapping class group of the space, which has been extensively studied. This talk will aim to give an overview of what is known in the higher-dimensional case. No prior knowledge of symplectic topology will be assumed.
BMC Morning Speaker
Matthew Keith (Strathclyde)
Analysis of Thin Leaky Dielectric Layers Subject to an Electric Field
The application of an electric field can have a significant effect on the behaviour of fluids. For example, electrohydrodynamic (EHD) instabilities can lead to droplet formation or nonlinear pattern formation which have an abundance of industrial applications such as inkjet printing and the production of micro-electronic devices. The recent review by Papageorgiou [1] gives an overview of the recent work on EHD instabilities. In the present work, we investigate a bilayer of liquid and gas contained between two solid walls subject to a normal electric field, adopting the full Taylor-Melcher leaky dielectric model [2]. This work builds on that of Wray et al. [3] who considered the enhancement and suppression of EHD instabilities, investigating the axisymmetric problem of a fluid layer coating the outside of a solid cylinder. Using the long-wave approximation, we explore the linear stability of the system. An investigation of the nonlinear regime highlights four characteristic behaviours of the system, namely, asymptotic thinning, contact with the upper wall, the return of the interface to its flat state, and singular touchdown behaviour. Numerically calculated plots of appropriate parameter planes are obtained. Of particular interest are the critical transitions between these characteristic states, which we investigate both analytically and numerically. Furthermore, we investigate the long-time behaviour in the asymptotic thinning regime and the onset of sliding, and explore the self-similar dynamics of the liquid-gas interface during touchdown and upper-wall contact. Finally, we also explore the physically relevant perfectly conducting limiting case which simplifies the system and allows for additional analytical and numerical progress.
[1] Papageorgiou, D.T., 2019. Film flows in the presence of electric fields. Annual Review of Fluid
Mechanics, 51, pp.155-187.
[2] Saville, D.A., 1997. Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annual Review of Fluid Mechanics, 29, pp.27-64.
[3] Wray, A.W., Papageorgiou, D.T. and Matar, O.K., 2013. Electrified coating flows on vertical
fibres: enhancement or suppression of interfacial dynamics. Journal of Fluid Mechanics, 735, pp.427-456.
CT02 Geophysical Fluid Dynamics
Amy Kent (Oxford)
Multiscale Mathematical Modelling for Tendon Tissue Engineering
Tendon cells respond to mechanical stimuli, meaning that careful regulation of imposed forcing is key to ensuring healthy tissue growth during tendon regeneration. One tissue engineering strategy is to seed cells on a biomaterial scaffold, then placed inside a bioreactor which controls the environment of the growing cells. We will introduce the Humanoid Robotic Bioreactor, under development by Pierre Mouthuy at the Nuffield Department of Orthopaedics, Rheumatology and Musculoskeletal Sciences, which applies physiologically-relevant loads to tendon cells growing on a fibrous scaffold. Understanding the relationship between stresses experienced by cells on the micro-scale, and forces imposed by the robot, will provide insight into fundamental tendon biology, inform operating regimes for tissue engineering and inform rehabilitation programs. Towards this end, we will present a model of fluid-structure interaction in a simplified scaffold geometry. Combining lubrication theory and homogenisation, this model provides a link between scales, using the solution of the cell-scale problem to inform parameters in the problem on scale of the scaffold. We will present simulations of candidate forcing regimes and discuss the implications for the mechanical environment experienced by cells in the bioreactor scaffold. Authors - Amy Kent (Oxford), James Oliver (Oxford), Sarah Waters (Oxford) , Pierre Mouthuy (Oxford) and Jon Chapman (Oxford).
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Oliver Kerr (City, London)
Double-diffusive instabilities at a suddenly heated sidewall
When a large body of salt-stratified fluid is heated from the side at a vertical boundary it causes the fluid near to the wall to rise. This results in the fluid near the wall being both warmer and saltier than the fluid at the same level further from the wall, giving rise to horizontal temperature and salinity gradients as well as vertical shear. If the heating is strong enough then instabilities can form. These have been observed experimentally.
A previous criterion for instabilities when the salinity gradient is strong and there is a relatively gradual increase in the wall temperature was determined by a quasi-static stability analysis. However, for weaker salinity gradients and/or faster heating the background state is intrinsically unsteady at the time of instability, and the quasi-static assumption is no longer valid and a stability analysis has been absent. We will look at these instabilities using the approach of Kerr and Gumm (2017) in their investigation of heating of fluid at isolated boundaries. They found the optimal evolution of a quadratic energy-like measure of the amplitude of the instabilities. The choice of the measure is not predetermined, but selected to minimize this optimal growth. This approach has been used previously for investigating double-diffusive instabilities at a heated horizontal boundary. Here we will apply it to the double-diffusive sidewall problem.
We will show that there are transitions between three basic modes of instability: the small and large Prandtl number modes found previously for the purely thermal problem, and a double-diffusive mode. We also find that varying the time from which the start of the growth of the instabilities is measured can be significant, and by delaying this a significant enhancement of the growth may be observed in some cases.
We will compare our findings with the results of earlier experiments.
CT08 Industrial fluids
Nayab Khalid (St Andrews)
An infinite geometric presentation for Thompson's group $F$
In this talk, I will present the algorithms I developed during the course of my PhD to find an infinite presentation for Thompson's group $F$ which reflects the geometric structure of the unit interval. Time permitting, I will also discuss how this presentation could help us solve the rotation distance problem.
BMC09 Groups
Cassandra Khan (Edinburgh)
Temperature control of nematicon trajectories
When a light beam propagates through a nematic liquid crystal, it heats the material and the resulting change in temperature in turn affects the propagation of the light. The authors propose a theory that reduces the mathematical complexity using several physical assumptions, and gives remarkable agreement with experimental data. Using modulation theory, we develop a simple [(2+1)-dimensional] model to describe the synergy between the thermo-optical and reorientational responses of nematic liquid crystals to light beams to describe the routing of spatial optical solitary waves (nematicons) in such a uniaxial environment. Introducing several approximations based on the nonlocal physics of the material, we are able to predict the trajectories of nematicons and their angular steering with temperature, accounting for the energy exchange between the input beam and the medium through one-photon absorption. The theoretical results are then compared to experimental data from previous studies, showing excellent agreement
MS22 Theory and modelling of liquid crystalline fluids
Karima Khusnutdinova (Loughborough)
Dispersive hydrodynamics in an elastic rod
We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross-section within the scope of the general weakly-nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive the extended Boussinesq and Korteweg-de Vries - type equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of near-identity transformations. These solutions are compared with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement within the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to the waves of moderate amplitude) as a very good initial guess. In particular, we were able to observe a stably propagating "table-top" soliton, well-known in the context of internal waves in stratified fluids.
Reference:
F.E. Garbuzov, Y.M. Beltukov, K.R. Khusnutdinova, Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and cubic nonlinearities, Theor. Math. Phys. 202 (2020) 319-333.
MS23 Dispersive hydrodynamics and applications
Minhyong Kim (Oxford)
Recent Progress on Diophantine Equations in Two Variables
The study of rational or integral solutions to polynomial equations f(x_1, x_2,.., x_n)=0 is among the oldest subjects in mathematics. After a brief description of its modern history, we will review a few of the breakthroughs of the last few decades and some recent geometric approaches to describing sets of solutions when the number of variables is 2.
BMC Morning Speaker
Eun-jin Kim (Coventry)
Information Geometry in non-equilibrium processes
A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability. This metric structure then provides a key link between stochastic systems and geometry. We define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time, and is called the information length. We apply this method to atmospheric data obtained from the global circulation model as well as the self-organising process in laboratory plasmas. In particular, we show that time-dependent PDFs are non-Gaussian in general, and the information length calculated from these PDFs shed us a new perspective of understanding variabilities, correlation among different variables and regions.
CT 09 Mathematical physics
Timothy King (King's College London)
Curved Schemes for SDEs on Manifolds
Given a stochastic differential equation (SDE) in Rn whose solution is constrained to lie in some manifold M ⊂ R n , we propose a class of numerical schemes for the SDE whose iterates remain close to M to high order. Our schemes are geometrically invariant, and can be chosen to give perfect solutions for any SDE which is diffeomorphic to n-dimensional Brownian motion. Unlike projection-based methods, our schemes may be implemented without explicit knowledge of M. Our approach does not require simulating any iterated Itˆo interals beyond those needed to implement the Euler–Maryuama scheme. We state a result on the convergence of the scheme, and give some experiments showing that the schemes perform well in practice; these include the stochastic Duffing oscillator, the stochastic Kepler problem, and also an application to Markov chain Monte Carlo.
Poster
Kristian Kiradjiev (Oxford)
Modelling removal of toxic gases using reactive filters
In the drive to protect the environment, reducing the concentrations of harmful chemicals that are released into the atmosphere has become a priority for industry. Many chemical filters contain reactive components where harmful substances are removed or transformed. In this talk, we derive a homogenised model for a flue-gas filter that converts sulphur dioxide into liquid sulphuric acid. We first consider a microscale porous domain, focused on a single catalytic pellet within the filter material, and derive the governing equations. We then homogenise over both the gaseous and the liquid phase to obtain macroscale equations for the concentration of sulphur dioxide and the thickness of the liquid sulphuric acid layer that grows around the pellets. There are two key dimensionless parameters that emerge as part of the analysis and govern the behaviour of the system, namely, the reaction rate at the pellet surface and the mass transfer across the gas--liquid interface. To obtain a complete model of the filter, we couple the macroscale equations, valid within the filter material, to an equation governing the external gas flow through the filter. We solve the resulting model and consider asymptotic reductions based on the filter geometry. In one distinguished limit we consider, we obtain an explicit solution for the sulphur dioxide concentration and the void fraction in the filter. We vary parameters such as the gas speed and establish the operating regimes for effective cleansing of flue gas. The model we develop retains generality and can be applied to other physical and industrial processes, where other toxic gases are involved.
MS29 IMA Lighthill Thwaites
Anna Kirpichnikova (The University of Stirling)
Construction of artificial point sources for linear wave equation and application for inverse problems
We study the wave equation on a bounded domain $R^m$ or on a compact Riemannian manifold with boundary.
Let us assume that we do not know the coefficients of the wave equation but are only given the hyperbolic Neumann-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. We show that it is possible to construct a sequence of Neumann boundary values so that at a time $t_0$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point where the energy of wave focuses is known in travel time coordinates, and satisfies a certain geometrical condition.
CT12 Waves
Rachel Kirsch (ISU)
Universal partial words
Chen, Kitaev, Mütze, and Sun recently introduced the notion of universal partial words, a generalization of universal words and de Bruijn sequences. Universal partial words allow for a wild-card character $\diamond$, which is a placeholder for any letter in the alphabet. For non-binary alphabets, we show that universal partial words have periodic $\diamond$ structure and are cyclic, and we give number-theoretic conditions on the existence of universal partial words. In addition, we provide an explicit construction for an infinite family of universal partial words over non-binary alphabets. Finally, we give a preliminary report on ongoing work.
Based on joint work with Bennet Goeckner, Corbin Groothuis, Cyrus Hettle, Brian Kell, Pamela Kirkpatrick, and Ryan Solava.
BMC06 Combinatorics
Rainer Klages (QMUL)
A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics
Consider equations of motion that generate dispersion of an ensemble of particles in the long time limit. An interesting problem is to predict the diffusive properties of such a dynamical system, starting from first principles. Here we consider an interval exchange transformation, lifted onto the whole real line, that is not chaotic, in the sense of exhibiting a vanishing Lyapunov exponent. We show analytically that this map nevertheless displays the whole spectrum of normal and anomalous diffusion under variation of a single control parameter. The propagating fronts generated in the superdiffusive regime bear some similarity with the ones obtained from stochastic Levy walks.
Joint work with Lucia Salari, Lamberto Rondoni and Claudio Giberti.
MS12 Front Propagation in PDE, probability and applications
Stefan Klus (Surrey)
Kernel methods for detecting coherent structures
Over the last years, numerical methods for the analysis of large data sets have gained a lot of attention. Recently, different purely data-driven methods have been proposed which enable the user to extract relevant information about the global behavior of the underlying dynamical system, to identify low-order dynamics, and to compute finite-dimensional approximations of transfer operators associated with the system. However, due to the curse of dimensionality, analyzing high-dimensional systems is often infeasible using conventional methods since the amount of memory required to compute and store the results grows exponentially with the size of the system. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics, fluid dynamics, and quantum mechanics.
MS09 Integrating dynamical systems with data driven methods
Alexander Koch (Manchester)
Mathematical modelling the circadian clock
The circadian clock, colloquilly known as the ‘body clock’, is a biomolecular ocillator intrinsic to all cells within the body which maintains temporal organisation throughout the body. With or without entrainment to external day/night rhythms the circadian network of genes, mRNAs and proteins autonomously osciallates with a 24-hour cylce. Despite qualitative knowledge via perturabative experiments it remains unknown how the collective feedback loops and molecular interactions of the mammalian clock quantitatively generates a daily cycle.
We are utilising new approaches in live cell imaging of fluorescent fusion proteins to quantify dynamics and interactions. Here, Fluorescence Correlation Spectroscopy (FCS) is used to measure the exact levels and interactions between fluorescent fusions of key proteins, including CLOCK, BMAL1 (activators), CRYPTOCHROMEs and PERIOD (repressors) expressed either constitutively (lentivirus) or endogenously (CRISPR). Strong interactions have been measured through fluroescence cross-correlation spectroscopy (FCCS) between CRYs and PERs flurorescently labelled in different colours. We are now determining an accurate value of the disassociation constant through novel bayesian inference of raw fluorescence data.
These experimental data alongside mathematical modelling are used to probe the relative importance of each component and motif in conferring robust timekeeping. Self-sustained single cell oscillations are modelled as a network of stochastic reactions with an inherent limit cycle. Of interest is the proximity of the limit cycle to its point of instability and hence robustness against molecular noise. In the lab we have measured the abundance of mRNA and proteins to be relatively small when compared to other molecular oscillators, in the range of 10s and 1000s respectively. In this regime noise is expected to contribute significantly and thus we wish to explore how robust timekeeping still occurs.
The comprehensive regularisation scheme informed by experimental data sets limits on the model and provides a method of quantifying uncertainty upon predictions. This is crucial to avoiding erroneous out of sample modelling predictions. These approaches will be one of the first demonstrations of quantitative live-cell measurements of clock interactions and provides insight into their role in the generation of cellular circadian rhythms.
Poster
Nickolay Korabel (Manchester)
Non-homologues repair process driven by anomalous dynamics of DNA double strand breaks
The end joining process during the nonhomologous repair of the DNA double strand breaks after radiation damage is considered. It has been experimentally established that double strand break ends do not follow simple diffusion but move subdiffusively. However, it is still an open question which subdiffusion model is more appropriate. To decipher the mechanism of sub diffusion, we extend the DNA Mechanistic Repair Simulator (DaMaRiS) developed recently by implementing two additional sub diffusion models, fractional Brownian motion and a fractional Langevin equation, to compliment theContinuous Time Random Walk model already present. We compare predictions of biologically measurable end points such as repair protein recruitment kinetics
and DSB repair kinetics for each sub diffusion model with published experimental data. We further calculate the probabilities of formation of a dicentric chromosome and an acentric chromosome fragment as a result of misrepair of DNA strand breaks.
CT10 Mathematical Biology-2
Theodoros Kouloukas (Kent)
Cluster maps associated with discrete KdV equation
In the context of cluster algebras, nonlinear recurrences are generated from cluster mutation-periodic quivers, i.e. quivers with certain periodicity property under sequences of mutations. In this talk we focus on a particular class of cluster recurrences associated with the discrete KdV equation which arise as plane wave reductions of the Hirota-Miwa (discrete KP) equation. We study the integrability of the corresponding discrete systems using the properties of the underlying cluster algebra structure.
BMC05 Math. Physics
Rosa Kowalewski (Bath)
Euler-Poincaré equations for nonconservative fluid dynamics
Dating back to W.R. Hamilton, the dynamics of a physical system are captured in an action functional, which by a variational principle yields the equations of motion of the system. The equations of motion, Euler-Lagrange equations, can be further reduced to the Euler-Poincaré equations if the system has an underlying symmetry.
In contrast to the Euler-Lagrange equations, which are expressed in a particular coordinate system in Eulerian (spatial) coordinates, the Euler-Poincaré equations are formulated in the Lagrangian (material) reference frame and can therefore be written without the use of a particular coordinate system.
There are physical laws which can not be captured by the traditional Hamilton’s principle: if the system involves non-conservative components the necessary time-symmetry is broken and Hamilton’s principle is not valid. A recently developed formalism by Galley (2012) allows the formulation of a variational principle for non-conservative systems, on an action functional of the doubled set of degrees of freedom. A 'potential' function, which couples the doubled variables, includes nonconservative interactions in the Lagrangian.
In this poster, we derive the Euler-Poincaré equations following from Galley's action principle for nonconservative fluid dynamics. In order to generalise the formalism to coordinate-free expressions, and to obtain deeper insight in the underlying geometry, we reformulate the principle in terms of deformations acting on the fluid manifold.
Poster
Vladimir Krajnak (Bristol)
Revealing phase space structures in the presence of high instabilities
Investigations of transport in Hamiltonian systems require the identification of invariant phase space structures that govern the dynamics. These phase space structures are the stable and unstable invariant manifolds of NHIMs (unstable periodic orbits for systems with 2 degrees of freedom). We introduce a method [5] for approximating individual branches of stable and unstable invariant manifolds of (but not limited to) highly unstable periodic orbits. The method enabled the investigation of roaming reaction dynamics [1,3,7] in Chesnavich's CH4+ model subject to the Hamiltonian isokinetic thermostat [2,6]. Certain of the periodic orbits that govern roaming in the system [4] have Lyapunov exponents ~10^21.
[1] J. M. Bowman and B. C. Shepler. Roaming radicals. Annu. Rev. Phys. Chem., 62:531–553, 2011.
[2] C. P. Dettmann and G. P. Morriss. Hamiltonian formulation of the Gaussian isokinetic thermostat. Phys. Rev. E, 54, 1996.
[3] V. Krajňák and H. Waalkens. The phase space geometry underlying roaming reaction dynamics. J. Math. Chem., 56, 2018.
[4] V. Krajňák, G. S. Ezra, and S. Wiggins. Roaming at constant kinetic energy: Chesnavich’s model and the Hamiltonian isokinetic thermostat. Regul. Chaotic Dyn., 24 2019.
[5] V. Krajňák, G. S. Ezra, and S. Wiggins. Using Lagrangian descriptors to uncover invariant structures in Chesnavich's Isokinetic Model with application to roaming. accepted by Int. J. Bifurcation Chaos, 2020.
[6] G. P. Morriss and C. P. Dettmann. Thermostats: Analysis and application. Chaos, 8, 1998.
[7] R. D. van Zee, M. F. Foltz, and C. B. Moore. Evidence for a second molecular channel in the fragmentation of formaldehyde. J. Chem. Phys., 99, 1993.
CT13 Dynamical Systems
Andrew Krause (Oxford)
Pattern Formation in Heterogeneous Environments: Turing's Theory of Successive Patterning
Motivated by experimental work on multiscale patterns in biology, I will discuss recent progress extending Turing's theory of diffusion-driven morphogenesis to the case of a spatially heterogeneous domain. We employed a WKBJ ansatz to study the emergence of Turing-type instabilities from a spatially heterogeneous steady state in regions which locally satisfy conditions for Turing instability, permitting pattern formation in an intuitive way across a spatially-varying domain. Such an extended theory allows us to account for successive patterning events during development, explaining microstructures within periodic patterns such as on the skin of jaguars. Our theory separates mechanisms of patterning due to environemntal forces from those related to population interactions, and hence may also be valuable for elucidating the causes of colony formation and niche partitioning in complex ecosystems.
CT10 Mathematical Biology-2
Hannah Kreczak (Newcastle)
Subsurface dynamics of biofouled microplastic
The presence of macro and micro plastic in the marine environment is a growing problem and there is need to understand the distribution of plastic within oceans to fully assess its impact. Biofouling, the accumulation of organisms on wetted surfaces, is a mechanism which has been observed to displace buoyant microplastics from the ocean surface. Observing the transport of plastic subsurface is difficult and so there is growing interest in developing deterministic models to predict plastic distributions. We present the results of a dynamical system coupling particle hydrodynamics and algal growth dynamics to capture the vertical transport of biofouled microplastic. Non-dimensionalisation of the system reveals scaling relations for particle resurface time and the maximum depth reached. Variations in ocean hydrography, such as sharp increases in fluid density and increased background algal concentration, effect the long-time motion of particles and we predict the presence of a subsurface plastic trapping layer centred around the euphotic zone depth.
Poster
Jan-Ulrich Kreft (Birmingham)
Predicting antibiotic resistance selective windows in wastewater treatment plants
Wastewater treatment plants receive a host of antimicrobial resistance gene carrying plasmids and bacteria. Despite strong reduction in the levels of resistance, the effluent still contains enough resistance genes to increase these in downstream sediments. We aim to predict the fate of resistance plasmids in activated sludge. We have developed an ordinary differential equation model based on the industry standard Activated Sludge Model 1 (ASM1) by adding three entities: (i) enteric bacteria, (ii) a resistance plasmid (with fitness cost) and hence splitting the bacterial populations into 4 groups (Recipients, Non-Recipients, Donors, Transconjugants) and (iii) an antimicrobial (that is degraded). This ASM1+++ model predicts how the minimal and maximal selective concentrations depend on environmental conditions. We find that costly plasmids can be maintained without antibiotic selection if the sewage is concentrated enough or if the transfer rate of the plasmid is boosted in transconjugants. If these conditions are not met, antibiotics can select for the presence of the plasmid at concentrations that are much lower than the rule of Minimum Inhibitory Concentration (MIC)/10 would suggest. In fact, the MIC is a poor predictor as the selective window depends more directly on the half maximal effective concentration (EC50) and also on a number of other factors such as mortality and solid retention time. In conclusion, the MIC is a poor basis for deriving selective windows and a more complete account of environmental factors has to be included. Minimal selective concentrations may be much lower than MIC/10 but still higher than concentrations typically found in wastewater treatment.
MS20 Mathematics of the water, energy and food security nexus
Daniela Kuhn (Birmingham)
Proof of the Erdos-Faber-Lovasz conjecture
The Erdos-Faber-Lovasz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. (Here the chromatic index of a hypergraph H is the smallest number of colours needed to colour the edges of H so that any two edges that share a vertex have different colours.) Erdos considered this to be one of his three most favorite combinatorial problems and offered $500 for the solution of the problem.
In joint work with Dong-yeap Kang, Tom Kelly, Abhishek Methuku and Deryk Osthus, we prove this conjecture for every large n. We also provide `stability versions' of this result, which confirm a prediction of Kahn.
In my talk, I will discuss some background, some of the ideas behind the proof as well as some related open problems.
BMC Morning Speaker
Vitaliy Kurlin (Liverpool)
Introduction to Periodic Geometry for materials applications
The Crystal Structure Prediction (CSP) aims to discover new solid crystalline materials (crystals) with desired properties for a given chemical composition. The typical approach to CSP is an almost random initial sampling of simulated crystals and their subsequent time-consuming optimization by supercomputers. The resulting CSP landscapes are unstructured plots of thousands or even millions of (often nearly identical) approximations to local minima. The main bottleneck in the CSP is the ambiguity challenge meaning that a real object such as a crystal can be represented in infinitely many different ways. Hence many similar crystals are treated as different and even more resources are wasted on running predictions of physical properties for near duplicate crystals. We introduce key concepts and first results in the new area of Periodic Geometry, which will enable a guided exploration in the space of all potential materials instead of the current random sampling.
MS11 Mathematics for Materials Science
Jochen Kursawe (St Andrews)
Quantitative models of gene expression dynamics during embryonic cell fate decisions
Understanding and regulating cell fate decisions will be crucial in many bio-medical applications, for example the growth of artificial organs or stem cell therapies. Traditionally, cell states are identified and described by investigating static gene expression profiles. However, in recent years it has become increasingly clear that dynamic patterns of gene expression, such as oscillations, can play important roles in cellular decision making. For example, gene expression oscillations have been proposed to control the timing of cell differentiation during embryonic neurogenesis, i. e. the generation of nerve cells. The mathematical analysis of gene expression dynamics may be hindered by sparse data and parameter uncertainty. Here, we combine Bayesian inference and quantitative experimental data on mouse and zebrafish neurogenesis to explore mechanisms controlling aperiodic and oscillatory gene expression dynamics during cell differentiation. We find that quantitatively accurate model predictions are possible despite high parameter uncertainty. We identify examples of stochastic amplification, where oscillations are enhanced by intrinsic noise and we show how such oscillations can be initiated by changes in biophysical parameters. We further consider mechanisms that may enable the down-stream interpretation of dynamic gene expression. Our analysis illustrates how quantitative modelling can help unravel fundamental mechanisms of dynamic gene regulation.
CT 16 Mathematical Biology-3
Matthias Kurzke (Nottingham)
The effect of forest dislocations on the evolution of a phase-field model for plastic slip
We consider dynamics of a phase-field model for crystal dislocations in the large body/small Burgers vector limit. In the one-dimensional Peierls-Nabarro setting without a forest dislocation background, the limit of the gradient flows of the energies is the gradient flow of the $\Gamma$-limit, similar to related problems in ferromagnetic materials. Forest dislocations introduce an extra strange term into the $\Gamma$-limit. Although this term may speed up the evolution of the $\Gamma$-limit, we show that it does not represent an additional driving force: instead, the presence of forest dislocations introduces a wiggliness into the system that actually slows down the observed evolution.
MS12 Front Propagation in PDE, probability and applications
Halim Kusumaatmaja (Durham)
Phase transitions on non-uniform curved surfaces: Coupling between phase and location
Confinement to a surface of varying curvature has a dramatic effect on the structure, kinetics and thermodynamics of clusters of attractive colloids. Using a combination of Monte Carlo, molecular dynamics and basin-hopping methods, we show that the stable states are distinguished not only by the phase of matter but also by their location on the surface. Furthermore, the transitions between these states involve cooperative migration of the colloidal assembly. These phenomena are general, and here they are explicitly demonstrated on both toroidal and sinusoidal surfaces for colloidal particles interacting with different ranges of Morse potential.
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Lisa Lamberti (ETH Zurich)
Cluster partitions and fitness landscapes of the Drosophila fly microbiome
In applications often data come as high-dimensional point configurations. Properties of such point configurations can be studied via subdivisions of convex polytopes. In this talk, I present how recent advancements in this theory help uncover new biological insights focusing on the case of experimental Drosophila microbiome data. This talk is based on joined work with Eble, Joswig and Ludington arXiv:2009.12277.
MS07 Applied Algebra and Geometry
Jake Langham (Bristol)
Stability of steady erosive shallow flows
All geophysical flows deposit and erode sediment to and from the environment. Strong and heavily sediment-laden flows such as volcanic lahars represent dangerous natural hazards; predictive physical models play a key role in understanding these flows and mitigating their effects. We present stability analyses of steady sediment-laden flows over constant grade, in a general shallow-water formulation with bed transfer terms. A key finding is that a diffusive term is always required to regularise these models (which otherwise become ill-posed at unit Froude number, leading to an inability to compute solutions under general initial and boundary conditions). Given sensible generic model closures, we observe two coexistent steady solutions (one stable, one unstable) with different sediment concentrations and discuss the geophysical implications of such states.
CT02 Geophysical Fluid Dynamics
Christopher Lanyon (Nottingham)
A model to investigate the impact of farm practice on antimicrobial resistance in UK Dairy Farms.
Dairy farm slurry tanks are repositories of solid and liquid bovine waste as well as farm waste and antibiotic contaminated milk. The antimicrobial resistance genes, antimicrobial resistant microbes and antibiotics discharged by cows, combined with the natural presence of bacteria, mean that the slurry tank is potentially a site for the development and proliferation of AMR. Slurry is spread onto crops as a fertiliser, making dairy farms a potential source of AMR to the environment.
Mathematical modelling can be used to quickly and efficiently identify important process factors using computer simulation. Particularly in the case of slurry storage, where practical experiments may conflict with a farmer's needs, models can be used to simulate changes in farm practice over a number of years without having to enact those changes.
To assess the impact of farm practice on the development of AMR we have designed a new ordinary differential equation (ODE) model of the prevalence and spread of AMR within the dairy slurry tank environment. We model the chemical fate of bacteriolytic and bacteriostatic antibiotics within the slurry and their effect on a population of Escherichia coli (E.coli) bacteria, which are capable of resistance to both types of antibiotic. Through our analysis we find that changing the rate at which a slurry tank is emptied may delay the proliferation of multidrug-resistant bacteria by up to five years depending on conditions. This finding has implications for farming practice and the policies that influence waste management practices. We also find that, within our model, the development of multidrug-resistance is particularly sensitive to the use of bacteriolytic antibiotics, rather than bacteriostatic antibiotics, and this may be cause for controlling the usage of bacteriolytic antibiotics in agriculture.
Our model demonstrates the influence of farm practices on the development and spread of AMR within the slurry tank. The model itself has implications outside of the UK, especially in countries such as America and China, where intensive dairy farming operations are particularly prevalent. This work will cover the mathematical development of the model, including our approaches to each of the dynamics which affect the prevalence and spread of antimicrobial resistance within the E.coli community. This will be followed by a discussion of our model analysis and recommendations for future research and data collection.
CT04 Mathematical Biology-1
Robert Laugwitz (Nottingham)
Construction of non-semisimple modular tensor categories using relative centers
Modular tensor categories are at the core of constructions of 3D topological field theories. These modular categories are assumed to be semisimple. More recently, there has been significant progress on non-semisimple generalizations of several applications of modular categories. In this talk, I will report on recent work with Chelsea Walton (Rice University) on using relative centers and double constructions in order to obtain example of non-semisimple modular categories.
Poster
Cindy Lawrence, Tim Nissen (MoMath)
National Museum of Mathematics (MoMath), New York
Mathematics illuminates the patterns that abound in our world. The National Museum of Mathematics strives to enhance public understanding and perception of mathematics. Its dynamic exhibits and programs stimulate inquiry, spark curiosity, and reveal the wonders of mathematics. The Museum’s activities lead a broad and diverse audience to understand the evolving, creative, human, and aesthetic nature of mathematics.
The National Museum of Mathematics began in response to the closing of a small museum of mathematics on Long Island, the Goudreau Museum. A group of interested parties (the “working group”) met in August 2008 to explore the creation of a new museum of mathematics — one that would go well beyond the Goudreau in both its scope and methodology. The group quickly discovered that there was no museum of mathematics in the United States, and yet there was incredible demand for hands-on math programming.
Accomplishments to date include: opening Manhattan’s only hands-on science center, welcoming more than one million visitors; creating the popular Math Midway exhibition, which has delighted millions of visitors at museums throughout the United States and internationally; leading math tours in various U.S. cities; running dozens of Math Encounters and Family Fridays events; delivering a broad array of diverse and engaging programs for students, teachers, and the public to increase appreciation of mathematics; and creating the largest public outdoor demonstration of the Pythagorean Theorem ever.
Further details: momath.org/
Outreach Video
Jack Lee (Kings College London)
Multiscale modelling of coronary circulation
Ischaemic heart disease continues to be a leading cause of death globally. Although the survival rate of acute coronary syndrome has improved, reduced myocardial blood flow (MBF) is associated with adverse remodelling, leading eventually to HFrEF. Impairment of coronary microcirculation has also been linked to the development of HFpEF, through disruption of intramyocardial signaling. Diagnosing and treating these disease requires an understanding of the blood flow in the whole heart across the large and small vessels.
Quantitative characterisation of MBF is a challenging problem and no existing imaging modality is able to capture its vast complexity at different scales. We propose multiscale modelling as a complementary means by which to estimate unobservable physiology in a rational manner. Over the past years we have developed a multiscale whole-heart cardiac perfusion model that couples discrete vessel flow models with a poroelastic contracting myocardium. The unique challenges this process entails, including the characterisation of vascular anatomy, constitutive laws and parameters, and strategies for coupling between scales will be examined. In particular, the difficulties in defining an optimal division between scales, and designing a corresponding coupling strategy, are highlighted as fundamental remaining modelling questions. Partial validation of the models may rely on coronary wave intensity analysis and perfusion magnetic resonance imaging. Our more recent work on robust processing of these data types (adaptive Savitzkey-Golay, Hierarchical Bayesian inference) will also be discussed.
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Joseph Leedale (Liverpool)
Multiscale modelling of drug transport and metabolism in liver spheroids
In early preclinical drug development, potential candidates are tested in the laboratory using isolated cells. These in vitro experiments traditionally involve cells cultured in a two-dimensional monolayer environment. However, cells cultured in three-dimensional spheroid systems have been shown to more closely resemble the functionality and morphology of cells in vivo. While the increasing usage of hepatic spheroid cultures allows for more relevant experimentation in a more realistic biological environment, the underlying physical processes of drug transport, uptake and metabolism contributing to the spatial distribution of drugs in these spheroids remain poorly understood. The development of a multiscale mathematical modelling framework describing the spatiotemporal dynamics of drugs in multicellular environments enables mechanistic insight into the behaviour of these systems. Here, our analysis of cell membrane permeation and porosity throughout the spheroid reveals the impact of these properties on drug penetration, with maximal disparity between zonal metabolism rates occurring for drugs of intermediate lipophilicity. Our research shows how mathematical models can be used to simulate the activity and transport of drugs in hepatic spheroids, and in principle any organoid, with the ultimate aim of better informing experimentalists on how to regulate dosing and culture conditions to more effectively optimise drug delivery.
CT10 Mathematical Biology-2
Marius Leonhardt (Heidelberg)
Plectic Galois action on Hilbert modular varieties
The modular curve has had various applications to number theory, in particular to the theory of elliptic curves. But what about applications of other Shimura varieties to higher dimensional abelian varieties? In this talk, we will focus on the Hilbert modular variety and explain one attempt to achieve such analogous applications, via the so-called "plectic conjecture" of Nekovar--Scholl. It involves a mysterious plectic Galois group and actions of this group on various quantities associated to the Hilbert modular variety. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible.
BMC02 Number Theory
Guoyang Li (Harvard)
Probing mechanical stress in soft materials with shear wave elastography
Quantitative mechanical characterization spans a wide spectrum of applications, from probing life sciences, visualizing health, to treating diseases. Shear wave elastography (SWE) is a fast-growing technique which enables us to quantitatively probe the mechanical properties of soft tissues on different length scales. The basic idea behind SWE is to utilize advanced imaging modalities, such as ultrafast ultrasound imaging and optical coherence tomography (OCT), to track the shear wave motion and solve an inverse problem based on the recorded wave motion to map the mechanical properties. Under physiological conditions, soft tissues (e.g., arterial wall, organs, cornea, etc.) may be subject to stress. While stress plays an important role in maintaining the mechanical homeostasis, evaluating the mechanical stress in soft tissues in vivo remains a challenge. It has been recognized that shear wave speed in load-bearing tissues varies with mechanical stress, a phenomenon called the acousto-elastic effect. Building on acousto-elastic theory, we recently proposed an ultrasonic method to measure stress in situ for loaded hard elastic solids such as steel. The superiority of this method is that no material constants need to be calibrated. Here we take a further step and measure mechanical stress in soft materials using SWE. This talk focuses on the principle of the proposed method and presents preliminary experimental results.
MS10 Ultrasonic Waves
Xin Li (Glasgow)
Interactions between C*-algebras, topological dynamics and group theory
C*-algebras are algebras of bounded linear operators on Hilbert spaces. Originally introduced as a mathematical foundation for quantum physics, these structures turn out to be interesting on their own right and exhibit a rich interplay with several other mathematical disciplines. Indeed, as I will explain, several key ideas which were initially developed to classify C*-algebras also lead to classification results for topological dynamical systems. At the same time, it was discovered recently that all C*-algebras which have been classified arise from dynamics in a precise sense. Surprisingly, this circle of ideas also leads to new constructions of groups which answered several open questions in group theory.
BMC Morning Speaker
Sauli Lindberg (Helsinki)
Taylor's conjecture on magnetic helicity conservation
Woltjer recognised magnetic helicity as a conserved quantity of ideal MHD. He proceeded to formulate Woltjer's variational principle which leads to the prediction of a relaxed state where the magnetic field is force-free. As later noted by Moffatt, ideal MHD also has many local conserved quantities (the subhelicities over magnetically closed subvolumes that move with the fluid). The relaxed state is nevertheless observed to be essentially independent of the local behaviour of the initial state.
A way out of the dilemma was conjectured by Taylor in 1974: in the presence of slight resistivity, the subhelicities cease to be conserved but the total magnetic helicity remains an approximate invariant. Berger showed in 1984 (under mild extra assumptions) that magnetic helicity dissipates much slower than magnetic energy. However, a rigorous mathematical proof straight from the MHD equations has been lacking. In mathematical terms, Taylor's conjecture translates into the statement that magnetic helicity is conserved in the ideal (inviscid, non-resistive) limit. I will discuss my recent proof, in collaboration with Daniel Faraco, of the mathematical version of Taylor's conjecture.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
Abigail Linton (NTNU)
Massey products in moment-angle complexes
Massey products are higher cohomology operations that are notoriously difficult to compute. Coming from Toric Topology, moment-angle complexes are spaces with many applications in commutative algebra, complex geometry and combinatorics. The topology of a moment-angle complex is often encoded combinatorially in its corresponding simplicial complex. I will demonstrate this by presenting systematic combinatorial constructions of non-trivial Massey products in moment-angle complexes. These constructions produce infinitely many families of manifolds with non-trivial Massey products and generalise all existing examples of Massey products in Toric Topology. This is joint work with Jelena Grbić (Southampton).
BMC03 Topology
Florian Litzinger (Queen Mary University of London)
Optimal regularity for Pfaffian systems and the fundamental theorem of surface theory
The fundamental theorem of surface theory asserts the existence of a surface immersion with prescribed first and second fundamental forms that satisfy the Gauss–Codazzi–Mainardi equations. Its proof is based on the solution of a Pfaffian system and an application of the Poincaré lemma. Consequently, the regularity of the resulting immersion crucially depends on the regularity of the solution of the corresponding Pfaffian system. This talk shall briefly review both the classical smooth case and the existing regularity theory and then introduce a recent extension to the optimal regularity.
BMC08 Analysis
Alan Logan (Heriot-Watt University)
Equalisers of free group homomorphisms and Post's correspondence problem
For two free group homomorphisms, we investigate the set of points where the two maps agree, called the equaliser of the maps, and we focus on two questions of Stallings from the 1980s about the bases of these equalisers.
There are two worlds from which to draw ideas. Firstly, fixed subgroups of free group automorphisms, which are particular instances of equalisers and which have generated a lot of literature since the 1980s. Secondly, equalisers of free monoid homomorphisms, which have been studied in computer science for over 70 years, starting with Post's correspondence problem, the (undecidable) decision problem asking `is the equaliser of two free monoid homomorphisms trivial?'.
Working somewhere between these worlds, we prove strong, positive results when both maps are `immersions' of free groups, and for the free group of rank two. Joint work with Laura Ciobanu.
BMC09 Groups
Sara Lombardo (Loughborough)
Integrability and plane wave instabilities: an algebraic-geometric approach
In this talk, I will review the approach to linear stability via integrability proposed in collaboration with Toni Degasperis and Matteo Sommacal (see Journal of Nonlinear Science, 28, pages 1251–1291, 2018). I will then apply it to study the linear stability of plane wave solutions of a novel long wave-short wave system which contains both the Yajima-Oikawa and Newell models for a particular choices of the coefficients. The stability spectra and the associated eigenfrequencies are explicitly computed, leading to a relation between the topology of the spectra and the gain function of the system.
Preliminary analysis indicates that, similarly to the case of vector Non-Linear Schroedinger (VNLS), the classification of the stability spectra allows one to predict regions of existence of rogue wave type solutions.
This work has been done in collaboration with Toni Degasperis, Matteo Sommacal and Marcos Caso Huerta.
BMC05 Math. Physics
Martin Lotz (Warwick)
Randomized dimensionality reduction in topological data analysis
Topological data analysis (TDA) is concerned with extracting topological features from discrete data. A common problem associated with methods for TDA is the high computational cost when dealing with large and high-dimensional data. We show that certain features of the data that are relevant for approaches to TDA such as persistent homology are preserved under random projections. Moreover, we show the target dimension does not depend on the size of the data but only on certain geometric parameters, and the projections can be computed efficiently. This gives a rigorous justification to the idea that the computational complexity of computations in TDA should only depend on the intrinsic dimension of the data, rather than the visible ambient dimension.
MS02 Mathematics for Data Science
Daniel Loughran (Bath)
Probabilistic Arithmetic Geometry
A theorem of Erdos-Kac states that the number of prime divisors of an integer behaves like a normal distribution (once suitably renormalised). In this talk I shall explain a version of this result for integer points on varieties. This is joint work with Efthymios Sofos and Daniel El-Baz.
BMC02 Number Theory
Nadia Loy (Torino)
Structure preserving schemes for Fokker-Planck equations with nonconstant diffusion matrices
In this talk we present an extension of a recently proposed structure preserving numerical scheme [1] for nonlinear Fokker-Planck-type equations to the case of nonconstant full diffusion matrices. While in existing works the schemes are formulated in a one-dimensional setting, here we consider exclusively the multi-dimensional case. We prove that the proposed schemes preserve fundamental structural properties like nonnegativity of the solution without restriction on the size of the mesh and entropy dissipation. Moreover, all the methods presented here are at least second order accurate in the transient regimes and arbitrarily high order for large times in the hypothesis in which the flux vanishes at the stationary state. Suitable numerical tests will confirm the theoretical results.
[1] L. Pareschi and M. Zanella. Structure preserving schemes for nonlinear Fokker-Planck equations and applications, J. Sci. Comput., 74(3):1575-1600, 2018.
[2] N. Loy and M. Zanella. Structure preserving schemes for Fokker-Planck equations with non-constant diffusion matrices. Submitted, 2019.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Ellen Luckins (Oxford)
Chemical reaction and counter-current flow in a silicon furnace
Industrial scale production of silicon involves reducing quartz rock (composed of silicon dioxide) with carbon in a submerged arc furnace. The chemical reactions converting these raw materials to silicon are highly endothermic; the heat energy required is provided both by radiation onto the surface of the raw materials, and by a flow of hot gas through the porous material bed. Although heat and mass transfer in the furnace depends on the chemical reactions, the interaction of these processes is not well understood. Motivated by this interesting industrial problem, in this talk we present a model for the counter-current heat and mass transfer between gases and the porous material bed in the furnace in the presence of an endothermic, temperature-dependent chemical reaction. Using the method of matched asymptotic expansions, we investigate various distinguished limits for different rates of heat transfer between the phases, assuming throughout that the effective Péclet number in the solid material is large. Through our analysis, we identify parameter regimes most applicable to the production of silicon, and employ our asymptotic solutions to provide new insights into those mechanisms underpinning the dynamics within a silicon furnace. Our theoretical analysis is of practical importance, in that it allows one to study dynamics within a furnace which are difficult to measure directly due to the extreme furnace temperatures. We conclude by discussing how our results are of use to our industrial partner, Elkem ASA, for improving the operation of real submerged arc furnaces.
MS29 IMA Lighthill Thwaites
Erwin Luesink (Imperial)
Stochastic variational principles for geophysical fluid dynamics
Recently, in [Holm2015] a stochastic variational principle was introduced to derive stochastic equations of motion for continuum mechanics. This development allows the introduction of data into models while preserving the underlying geometric structure. In [BdLHLT2020], such variational principles were put on some more rigorous footing, by proving that the constraint used to introduce the stochastic component is well-posed. In this talk I will explain why there is a need for stochastic variational principles, why they are of the form used in [Holm2015] and [BdLHLT2020] and how they are used.
[Holm2015]: Holm, D.D., 2015. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2176), p.20140963.
[BdLHLT2020]: de Léon, A.B., Holm, D.D., Luesink, E. et al. Implications of Kunita–Itô–Wentzell Formula for k-Forms in Stochastic Fluid Dynamics. J Nonlinear Sci (2020). https://doi.org/10.1007/s00332-020-09613-0
MS14 Variational Methods in Geophysical Fluid Dynamics
Xiaoyu Luo (Glasgow)
Growth and remodelling from the current configuration - application to the myocardium
Understanding growth and remodelling (G&R) of the myocardium is important for treatment and management of heart diseases. A commonly used approach for soft tissue G&R is the volumetric material growth, introduced in the framework of finite elasticity. In this approach, the total deformation gradient tensor is decomposed so that the elastic and growth tensors can be studied separately. A key element therefore is to determine the growth evolution. Most of existing volumetric growth theories assume that growth occurs from the natural (reference) configuration. In a few studies where the growth from the current configuration is considered, assumptions of compatible deformation or spherical elastic bodies are introduced. In reality, soft tissue G&R is continuous progress, so growth must occur from the current configuration. In this work, we seek to develop a framework of G&R from the current configuration, without releasing the residual stress or making any geometrical restrictions. We illustrate our idea using a simplified left ventricle model, which admits inhomogeneous growth in a residually-stressed current configuration. We then compare the residual stress distribution of different approaches with published experimental measurements and show that only the proposed framework can lead to a qualitative agreement.
Authors - Xin Zhuan (Glasgow), Ray Ogden (Glasgow), Xiaoyu Luo (Glasgow)
MS18 Growth and Remodelling in Soft Tissues
Frank Lutz (TU Berlin)
Reconstructing metallic foams from tomography data
Simple foams respect Plateau's rules combinatorially in the sense that every cell edge is contained in exactly three cells and every vertex is contained in exactly four cells. Though simple foams can be reconstructed from their adjacency graphs, in practice, the registration process of adjacency graphs from tomography greyscale image data of metallic foams comes along with errors. We discuss heuristics for the correction of the registration errors as a preprocessing step for the combinatorial analysis and roundness computation of the resulting foam structures. (Joint work with Ihab Sabik and Paul H. Kamm.)
MS11 Mathematics for Materials Science
Samantha Lycett (Roslin)
RNA virus strain diversity and fitness in meta-populations
RNA viruses have small genomes (5-30kb) which undergo rapid but error prone replication within a host, leading to within host diversification and the potential onward transmission of measurably
different viral variants. At the host population scale, and within and between hosts, the similar processes of viral diversification, selection of fit viral variants, infection between compartments and population bottlenecks occurs. These processes all leave signatures in the viral genomes, and
viral sequence data can be used with phylodynamic and phylogeographic methods for inferring viral effective population sizes over time, and spatial or between host transmission patterns.
Understanding the rise and spread of new variants is important for vaccine design in many animal and human disease systems, including SARS-CoV-2, where over the last year there has been a massive global sequencing effort. Here I will illustrate the use of phylodynamic and phylogeographic models for estimating the fitness of viral variants, and discuss how intrinsic viral properties, host immunity or host-population effects can influence what is observed and becomes dominant, using porcine reproductive and respiratory syndrome virus and SARS-CoV-2 as examples.
MS05 Multiscale Modelling of Infectious Diseases
Jane Lyle (Surrey)
Symmetric Projection Attractor Reconstruction: Multi-dimensional Embedding of Physiological Time Series
The proactive recording of many physiological signals, such as the electrocardiogram (ECG), is increasingly routine, generating large quantities of data, from which diagnostic and predictive information must be derived. Traditional analysis of such time series data tends to identify local extrema and intervals between them, and therefore discards the morphological detail of the whole waveform. Our aim is to capture the waveform shape information of non-stationary, approximately periodic signals as a bounded two-dimensional attractor to allow simple interpretation and quantification.
Takens' method of delay coordinates takes N points from a single continuous variable to reconstruct an attractor in an N-dimensional phase space. We have previously described our Symmetric Projection Attractor Reconstruction (SPAR) approach that adapts this to extract information from any approximately periodic waveform through an embedding in a three-dimensional phase space and subsequent projection to a two-dimensional image which can easily be quantified.
We present an extension of our SPAR method for embeddings in any dimension N2, whilst
retaining a two-dimensional output, and obtain a simple generalised result for a periodic signal. The derivation of this reveals that the two-dimensional attractor is a visualisation of the Fourier coefficients of the trigonometric interpolating polynomial of the embedding vector, which aids our understanding of the attractor properties, and its relationship to the underlying signal.
We then extend our generalised result into the space of approximately periodic signals. The ECG is a complicated signal reflecting various processes. Generating attractors in multiple higher dimensions provides information on subtle changes in different parts of the ECG morphology. We have previously applied machine learning to discriminate gender in the ECG, and we build on this to show that an individual's response to treatment with cardioactive drugs can be quantified from the trajectory of changes observed in the attractor.
The extension of the SPAR method into higher dimensions whilst retaining a two-dimensional projection has produced an interesting theoretical result which we have been able to apply and interpret in approximately periodic signals. In the clinical context, SPAR analysis of physiological signals has simple visual appeal, and provides support for a stratified approach to the diagnosis and management of patients.
CT 15 Statistical and Numerical Methods
Andre Macedo (Reading)
Local-global principles for norms
Given an extension L/K of number fields, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm everywhere locally is in fact a global norm from L. If L/K is cyclic, the original Hasse norm theorem states that the HNP holds. More generally, there is a cohomological description (due to Tate) of the obstruction to the HNP for Galois extensions.
In this talk, I will present work developing explicit methods to study this principle for non-Galois extensions. I will additionally discuss some recent generalizations of these methods to study the Hasse principle and weak approximation for products of norms as well as consequences in the statistics of these local-global principles.
BMC02 Number Theory
Fiona Macfarlane (St Andrews)
From a discrete model of chemotaxis with volume-filling to a generalised Patlak-Keller-Segel model
We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. We carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models.
CT10 Mathematical Biology-2
Antony Maciocia (Edinburgh)
An Update on Moduli of Sheaves and Bridgeland Stability Conditions
To a variety or space we can associate moduli spaces of sheaves. These moduli spaces have their own intrinsic interest as well as helping us to understand the geometry of the original space. They also arise in a number of other ways and playing constructions off against each other allows us to understand their structure better. One such way is through the very general notion of Bridgeland stability. In dimension two the situation is fairly well understood but very little is known in dimension three. In this talk I will describe some of the background ideas and demonstrate a few of the tools we are using currently to analyse moduli spaces and especially in dimension three.
BMC Morning Speaker
Diane Maclagan (Warwick)
Toric Bertini Theorems and Higher connectivity of Tropical Varieties
The classical Bertini theorem is a basic and fundamental tool in algebraic geometry. Recent work of Fuchs, Mantova, and Zannier extended this to a version where a hyperplane is replaced by a subtorus. I will discuss this result, and an application in tropical geometry (joint with Josephine Yu), where we use this to show that tropical varieties are highly connected.
BMC07 Algebraic Geometry
Cicely Macnamara (St Andrews)
Biomechanical modeling of cancer - Agent-based force-based modelling of solid tumours within the context of the tumour microenvironment
Once cancer is initiated, with normal cells mutated into malignant ones, a solid tumour grows, develops and spreads within its microenvironment invading the local tissue; the disease progresses and the cancer cells migrate around the body leading to metastasis, the formation of distant secondary tumours. Interactions between the tumour and its microenvironment drive this cascade of events which have devastating, if not fatal, consequences for the human host/patient. Among such interactions, biomechanical interactions are a vital component. In this talk several of these biomechanical interactions, often governed by repulsion and adhesion forces, will be discussed. A 3D individual-based force-based model, which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels, will be presented.
MS06 Tracking cellular processes through the scales
Elena Mäder-Baumdicker (Darmstadt)
The Morse Index of Willmore Spheres
Robert Bryant showed that any closed immersed Willmore sphere in Euclidean three-space is the inversion of a complete minimal sphere with embedded planar ends. We proved that the Willmore Morse Index of the closed surface can be computed by using unbounded Area-Jacobi fields of the related minimal surface. As a consequence, we get that all immersed Willmore spheres are unstable except for the round sphere. This talk is based on work with Jonas Hirsch and Rob Kusner.
BMC08 Analysis
Katie Madine (Liverpool)
Green's Functions in Discrete Flexural and Elastic Systems
The Green's function is the canonical object of study for many problems associated with wave propagation in structured solids; it contains the fundamental information of the dynamic response of the system. There has previously been a substantial amount of literature on Green's functions for scalar systems, such as photonics, acoustics, and platonics, particularly in statics. There has been comparatively little work on constructing Green's functions for dynamic vector systems corresponding to multi-scale mechanical materials. This area of research has applications in elastic metamaterials, non-destructive evaluation, one-way edge waves, cloaking, and seismic protection.
This talk discusses the unstudied problem of propagating flexural waves through discrete lattice systems. We construct the dynamic lattice Green's function for one- and two-dimensional flexural lattices. The dispersion equation is found explicitly, and then used to locate the band gaps of the system. The band gap Green's function for the vertical displacement motion of a localised defect mode is found explicitly for the system. The rotational effect on the masses at the connecting nodes as flexural waves travel down the system is also studied. The novel results presented lead to interesting phenomena, including localised defect states, dynamic anisotropy, and tensorial inertia matrices -- providing links to micropolar and Willis-type media.
CT12 Waves
Anotida Madzvamuse (Sussex)
A mechanobiochemical model for 3D cell migration
In this talk, I will review models for 3D cell migration and then present mechanobiochemical models that couple mechanical properties (viscoelastic) and biochemical processes (reaction-diffusion) to unravel mechanisms for 3D cell migration. The models are solved by use of the moving grid finite element method (which can be thought of as the bulk-surface finite element method). Numerical are exhibited to demonstrate different morphological dynamics of cells during migration.
MS06 Tracking cellular processes through the scales
Andrew Mair (Heriot-Watt)
Modelling the relationship between soil moisture and root density dynamics
Be it through carbon fixation or the provision of food and medicines, plants play a crucial role in supporting animal life on earth. Biological knowledge of the interactions between plants and their environment is rich and mathematical models of such interactions can provide valuable information regarding the effects of environmental stresses on plant life. In this project we derive and numerically simulate a model for the relationship between soil moisture and root density dynamics. The model consists of Richards' equation for water flow in soil, coupled with a system of transport equations describing root density dynamics within the soil.
There are many sources of empirical evidence which suggest that root density levels have a significant effect upon the hydraulic properties of soil. In our work we have used multi-scale analysis (homogenization) methods to suggest how the dependence on root density could be encoded into Richards' equation. Mathematical models for root density dynamics are usually transport equations involving both spatial and angular dimensions. We propose a root density model with only spatial dimensions so that, when coupled with Richards' equation, the system is numerically tractable.
The last section of the talk will describe the development of a numerical scheme to simulate our system of nonlinear partial differential equations. There are inherent challenges in numerically approximating a coupled system of Richards' equation and transport equations, the numerical methods employed to address these challenges will be discussed.
CT06 Porous media
Apala Majumdar (Strathclyde)
Pattern Formation in Confined Nematic Systems
Nematic liquid crystals are classical examples of partially ordered materials intermediate between isotropic liquids and crystalline solids. We study spatio-temporal pattern formation for nematic liquid crystals in two-dimensional regular polygons, subject to physically relevant non-trivial tangent boundary conditions, in the powerful continuum Landau-de Gennes framework. This is relevant to nematics on surfaces or confined within shallow three-dimensional wells. We study the qualitative properties of the stable equilibria as a function of the domain size, geometrical parameters, boundary conditions and material properties. In particular, we give novel insight into how we can tailor the structure, locations and multiplicity of defects by tuning the geometrical parameters. We also study saddle point solutions and transition states that connect distinct stable equilibria and propose new classification schemes for such unstable equilibria based on their Morse indices and number of interior and boundary defects. This is joint work with a large number of collaborators, all of whom will be acknowledged during the talk.
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Angelika Manhart (UCL)
Counter-propagating wave patterns in a swarm model with memory
Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. I will focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. Modelling how the behaviour depends on the time since the last reversal introduce a memory effect. The highlight consists of the analysis of counter-propagating travelling waves, patterns which have been observed in bacterial colonies: We find two families of interacting travelling waves whose discontinuous profiles remain unchanged, but whose composition is modified by the oncoming wave. I will discuss the explicit construction of such waves, show stability results and simulations.
MS12 Front Propagation in PDE, probability and applications
Angelika Manhart (UCL)
Emergence of macroscopic patterns due to microscopic swimmer-obstacle interactions
Aggregation of individuals, such as people, bacteria or sperm, is an ubiquitous phenomenon and is often attributed to direct attraction between the agents. In this talk I will present a basic model that suggests how aggregation can also be a consequence of interactions with an elastic environment.
I will start with a stochastic individual-based model (IBM) of collectively moving self-propelled swimmers and elastically tethered obstacles. Simulations reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. This motivates the derivation of a macroscopic partial differential equations (PDE) model for swimmer-obstacle interactions, for which we assume strong obstacle springs. The result is a coupled system of non-linear, non-local PDEs. Linear stability analysis allows to investigate pattern appearance and properties. Close inspection of the derived convolution operator in the PDE model reveals short-ranged swimmer aggregation, irrespective of whether obstacles and swimmers are attractive or repulsive.
MS06 Tracking cellular processes through the scales
Ciprian Manolescu (Stanford)
Khovanov homology and four-manifolds
Over the last forty years, most progress in four-dimensional topology came from gauge theory and related invariants. Khovanov homology is an invariant of knots in of a different kind: its construction is combinatorial, and connected to ideas from representation theory. There is hope that it can tell us more about smooth 4-manifolds; for example, Freedman, Gompf, Morrison and Walker suggested a strategy to disprove the smooth 4D Poincare conjecture using Rasmussen's invariant from Khovanov homology. It is yet unclear whether their strategy can work, and I will explain some of its challenges. I will also review other topological applications of Khovanov homology, with regard to smoothly embedded surfaces in 4-manifolds.
Plenary (LMS)
Ciprian Manolescu (Stanford)
Relative genus bounds in indefinite four-manifolds
Given a closed four-manifold X with an indefinite intersection form, we consider smoothly embedded surfaces in X - B^4, with boundary a knot K. We give several methods to produce bounds on the genus of such surfaces in a fixed homology class. Our techniques include relative adjunction inequalities and the 10/8 + 4 theorem. In particular, we present obstructions to a knot being H-slice (that is, bounding a null-homologous disk) in a four-manifold. We give an example showing that the set of H-slice knots can detect exotic smooth structures on closed 4-manifolds. Further, we give examples of knots that are topologically but not smoothly H-slice in some indefinite 4-manifolds. This is joint work with Marco Marengon and Lisa Piccirillo.
BMC03 Topology
Orlando Marigliano (KTH Stockholm)
Linear spaces of symmetric matrices
Linear spaces of symmetric matrices are interesting in a variety of mathematical contexts. For instance, they come up in optimization, algebraic geometry, and statistics. In the summer of 2020, about 40 researchers gathered on-line over several months to advance the understanding of these objects. In this talk, I report on some of the interesting results that came out of this effort. I also include my own contributions in the context of algebraic statistics. There, the linear spaces play the role of linear Gaussian concentration models. For these models, I discuss geometric and intersection-theoretic formulas for the maximum likelihood degree. I also discuss the more specific example of colored graphical models and their symmetries.
MS07 Applied Algebra and Geometry
Glenn Marrion (BioSS)
ABC-MBP: Better inference for national-scale epidemic models of SARS-CoV-2 transmission
This talk describes the development of practical inference methods for national-scale stochastic spatial models that can make use of a range of data of different types, resolutions and quality. Attempts to fit such models using current methods for inference, including data-augmentation and particle filtering, are briefly described before outlining a novel approximate algorithm that outperforms current methods. This new ABC-MBP algorithm combines benefits of standard Approximate Bayesian Computation and a data-augmentation approach based on model-based-proposals. Application to publicly available data, including analysis of commuting data collected during the 2011 U.K. census, enables estimation of key aspects of COVID-19 disease dynamics including transmission rates and the impacts of the first lockdown on the reproduction number.
MS28 Covid-19 Modelling
Rebecca McKinlay (Strathclyde)
Coating Flow on a Rotating Horizontal Circular Cylinder Subject to a Radial Electric Field
The two-dimensional dynamics of a thick film of an electrified, perfectly conducting Newtonian fluid flowing on the surface of a rotating horizontal cylinder are studied. The rotating cylinder is an electrode held at a constant potential and a concentric outer electrode whose potential is allowed to vary spatially encloses the system, inducing electrostatic forces at the interface. The long-wave approximation is used along with the method of weighted residuals to derive a model that incorporates the effects of the electric stress, rotation, gravity, viscosity, inertia, and capillarity. This model is investigated both numerically and analytically in appropriate limits to validate against known results. Novel second-order models governing the electric potential are derived by projecting onto asymptotically accurate polynomials and are validated against direct numerical simulations.
CT01 Viscous Fluid Dyamis 1
Elle Mclean (UCL)
Free Overfall Flow
Many works have considered two-dimensional free-surface flow over the edge of a plate, forming a waterfall, and with uniform horizontal flow far upstream. The flow is assumed to be steady and irrotational, whilst the fluid is assumed to be inviscid and incompressible, and gravity is taken into account. In particular, amongst these works, numerical solutions for both supercritical and subcritical flows are computed by Dias and Tuck (1991), utilising conformal mappings as well as a series truncation and collocation method. I will present an extension to this work where a more appropriate expression is taken for the assumed form of the complex velocity. The justification of this lies in the behaviour of the waterfall flow far downstream and how the parabolic nature of such a free-falling jet can be better encapsulated. New numerical results will be presented, demonstrating the difference in the shape of the new free surface profiles. Comparisons with the asymptotic solutions found by Clarke (1965) will also be made to validate these numerical solutions.
Poster
Andrew McLeod (Oxford)
Pyramid Ricci Flow
In joint work with Peter Topping we introduce pyramid Ricci flows, defined throughout uniform regions of spacetime that are not simply parabolic cylinders, and enjoying curvature estimates that are not required to remain spatially constant throughout the domain of definition. This weakened notion of Ricci flow may be run in situations ill-suited to the classical theory. As an application of pyramid Ricci flows, we obtain global regularity results for three-dimensional Ricci limit spaces (extending results of Miles Simon and Peter Topping) and for higher dimensional PIC1 limit spaces (extending not only the results of Richard Bamler, Esther Cabezas-Rivas and Burkhard Wilking, but also the subsequent refinements by Yi Lai).
BMC08 Analysis
Richard Mcnair (Manchester)
Amazing Marangoni: Maze solving with surfactant dynamics
Authors: Richard Mcnair, Oliver Jensen, Julien Landel
Experiments (Temprano-Coleto et al., Phys. Rev. Fluids 3:100507, 2018) have shown how exogenous surfactant introduced to the entrance of a maze filled with a shallow liquid layer will induce a flow in the liquid which spontaneously finds the longest path through the maze, effectively solving the maze with minimal flow into dead end sections of the maze. Here we test the hypothesis that this is due to the dynamics of endogenous contaminant surfactants already present in the liquid. We propose a model based on lubrication theory to derive equations capturing the Marangoni flow induced by the exogenous and endogenous surfactants at the film surface. The equations reduce to a nonlinear diffusion equation which describes the late-time spreading of the exogenous surfactant in the presence of endogenous surfactant. A further equation is needed to track the leading edge of the introduced exogenous surfactant. Numerical simulations of the equations describing the flow and exogenous surfactant front location qualitatively capture the experimental data. A numerical method for solving the nonlinear diffusion equation on a network using tools from graph theory and discrete calculus is also presented.
CT12 Waves
Gordon McNicol (Glasgow)
Self-excited oscillations in flow through a flexible-walled channel with a heavy wall
We develop a model for laminar highReynoldsnumber flow through a long finitelength planarchannel, where a segment of one wall is replaced by a membrane of finite mass that is held underlongitudinal tension. Driving the flow using an imposed upstream pressure, we employ a linearstability analysis to investigate the static and oscillatory global instabilities of the system, predicting thecritical flow conditions required for the onset of selfexcited oscillations. We show that the primaryglobal instability of the system involves an oscillating wall profile with a single extremum. We furthershow that increasing the wall mass and membrane tension have contrasting influences on theoscillation frequency of this mode. We examine the asymptotic behaviour of the primary global mode inthe limit where the wall mass and tension become large simultaneously and the oscillation frequencyremains finite. In this regime we develop analytical expressions for the critical Reynolds number andoscillation frequency, which show excellent agreement with numerical results. Furthermore, we use aweakly nonlinear analysis to elucidate the mechanism driving the instability: oscillation reduces themean flow rate along the channel, reducing the overall dissipation and requiring less work done byupstream pressure than in the basic state.
Poster
Kitty Meeks (Glasgow)
Graph Theory on the Farm
Graph Theory on the Farm is a collection of resources designed to introduce topics from discrete maths to groups of all ages through interactive farmyard-themed activities and games. In the video I discuss some of the motivation behind the concept, as well as my experiences of working with two undergraduate students to produce the resources. More information – and the resources themselves – are available at dcs.gla.ac.uk/~kitty/GTotF.
Outreach Video
Sumit Mehta (Hyderabad, India)
Buckling of hyperelastic circular plate due to growth
In this work, we have explored the growth-induced buckling in an isotropic incompressible hyperelastic circular plate with constraint boundary. Asymptotic finite-strain plate theory is used to investigate the large deformations in the plate. We first derived the 2-D plate equation from a 3-D governing system for plates using variational principle. The linear stability analysis is performed for the radially growing plate to investigate the buckling value of the growth factor. The effect of growth on the buckling behavior of the plate is analyzed for different thickness values. We observe that the finite-strain plate theory ensures the correct results for the 2nd order of thickness which are also compared with the results corresponds to O(h). These results are applicable to modeling growth induced deformation in soft biological tissues such as skin and in flexible electronics.
Poster
Brady Metherall (Oxford)
Clogging of Charge in Silicon Furnaces
High temperature gas created by chemical reactions involved in the releasing of silica in a furnace is vented up long narrow channels in a granular material. Cooling of the gas by heat loss to the channel walls causes condensation, which builds up naturally on the surfaces, and can clog the channels. Understanding this process is critical as clogged channels cause the pressure to increase excessively, which can lead to dangerous gas blows. A mathematical model is developed for the growth of brown condensate in idealized channels within the charge of industrial silicon furnaces. We derive governing equations for the temperature in the gas and condensate, the concentration of silicon monoxide and carbon monoxide, the fluid flow of the gas through the channel, and finally, conservation equations on the gas-condensate interface. We reduce this model after non-dimensionalization, and the resulting equations are solved numerically. We find that channels approximately 6mm in width accumulate the most condensate, however, the clogging time increases monotonically with width. The critical width corresponds to a Peclet number of order one.
Poster
Francesco Migliavacca (Milano)
The thrombectomy procedure: towards patient-specific modelling
An acute ischemic stroke appears when a blood clot (thrombus) in a cerebral artery prevents the blood to supply the downstream tissues. One of the available remedies for stroke patients is the intra-arterial thrombectomy, which is a minimally invasive procedure based on stent technology. After stent deployment, the thrombus is dislodged by retrieval of the stent. This study proposes a workflow to set-up and to perform an in silico thrombectomy procedure.
Patient-specific cerebral arteries were reconstructed by segmenting the images from contrast CT scans and they were discretized with quadrilateral rigid elements. Thrombi were drawn and positioned according to collected clinical data (Clot location and length, histology and Clot Burden Score), discretized with tetrahedral elements and modeled with compressible hyperelastic material. Parametric CAD geometries of three different commercial-available stent-retrievers were created with a Python script to replicate the real design of the main commercial-available stent-retrievers. They were discretized with beam elements and the Ni-Ti material was modeled as a shape memory alloy after a proper parameters calibration with coupled in vitro/in silico tensile tests. The thrombectomy simulation was set in accordance with the registered clinical data and performed with the explicit finite-element solver LS-DYNA (ANSYS).
The simulations showed that our models can replicate the thrombus extraction, even in the presence of a tortuous vessel or in case of unsuccessful thrombectomy, where the thrombus slides from the stent. Furthermore, the numerical models gave indications on the stresses and strains on the clot, which can be related to the potential fragmentation of the clots during the procedure.
The proposed workflow is amenable to model the thrombectomy procedure and it could be used to predict the procedure outcomes but also to optimize the procedure itself or the stent-retriever design.
MS27 Mathematical and Computational Modelling of Blood Flow
Paul Milewski (Bath)
Mode Two Solitary Waves in Stratified Flows
There is an extensive literature on modelling, computation and observation of horizontally propagating waves in stratified flows. The vast majority of this work, particularly when it concerns nonlinear structures and solitary waves, focuses on “mode one”, that is, the fastest wave in the system whereby all the pycnoclines are deflected with the same polarity. The simplest model for mode one waves is the two-layer flow of a lighter fluid above a heavier one bounded above and below by rigid boundaries. In that case the mode one wave is the only wave present, and, in the long wave limit, the KdV and mKdV arise as weakly nonlinear models, and MCC as a strongly nonlinear model. Mode two waves minimally require an additional layer in order for two interfaces to deflect with opposite polarity (mode two), or the same polarity (mode one). Mode two waves are increasingly believed to be of great scientific importance for their role in ocean transport. We shall consider the three-layer problem in this talk using KdV and MCC-like models, and the full Euler equations. We shall describe the problem and suggest an answer to the question: do mode two solitary waves exist in the Euler equations?
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Laura Miller (Glasgow)
Effective balance equations for poroelastic composites
We derive the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. We assume that the size of the pores (the microscale) is comparable with the distance between adjacent subphases and is much smaller than the size of the whole domain (the macroscale). We then decouple spatial scales embracing the asymptotic (periodic) homogenization technique to derive the new macroscale model by upscaling the fluid–structure interaction problem between the elastic constituents and the fluid phase. The resulting system of partial differential equations is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model, which are to be computed by solving appropriate cell problems which reflect the complexity of the given microstructure. The model reduces to the limit case of simple composites when there are no pores, and standard Biot’s poroelasticity whenever only the matrix–fluid interaction is considered. We further prove rigorous properties of the coefficients, namely (a) major and minor symmetries of the effective elasticity tensor, (b) positive definiteness of the resulting Biot’s modulus, and (c) analytical identities which allow us to define an effective Biot’s coefficient. This model is applicable when the interactions between multiple solid phases occur at the porescale, as in the case of various systems such as biological aggregates, constructs, bone, tendons, as well as rocks and soil.
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Frank Millward (Manchester)
Unsteady wind-blown volcanic plumes
Erupting volcanoes often release large quantities of ash and gas that rise in a hot plume through a stratified atmosphere, until the density of the plume and atmosphere are equal. At this neutral buoyancy level, the plume ceases to rise and begins to spread out around the neutral buoyancy layer as a turbulent intrusion. The spreading is predominantly driven by buoyancy forces, and resisted by drag from the surrounding moving ambient. We consider the case of intrusions released into a uniform wind field. Eruptions can be highly unsteady, occurring as a sequence of small pulses and forming several intrusions of finite volume as they are carried downstream by wind. Sustained eruptions emit a continuous flux of ash, forming a single intrusion of increasing volume. The manner of release influences the rate at which intrusions spread in the crosswind direction. The radius of an intrusion of finite volume increases with time like t to the power 2/9, while the width of continuously supplied intrusions grows as t to the power 1/3. It is therefore important to understand whether sporadic pulses will meet downstream to form continuous flux-like intrusions as it could result in an increased rate of spreading. Similarity solutions for cloud width based on shallow water models of these eruptions are used here to predict whether initially distinct intrusions can merge downwind to form continuously supplied intrusions. Our theoretical results are then compared with both numerical models of intrusions and satellite data from the Raikoki eruption in June 2019.
Poster
Matthias Mimault (James Hutton I)
Smoothed Particle Hydrodynamics Method to model root development
Plant organs develop from a group of undifferentiated cells called the meristem. Meristematic cells operate sequences of elongation, division and differentiation from which emerge complex morphologies and tissue architectures. The kinematics of development in plants is well characterised but understanding the coordination of cell activity is challenging. Indeed, a mathematical framework encompassing the discrete nature of cellular processes while solving biophysical dynamics at the macroscopic scale remains out of reach.
Here, we propose a cell-based model capable to simulate entire root meristems. The model is based on the Smoothed Particle Hydrodynamics (SPH) method and allows to solve equations for the growth at tissue scale from a flexible distribution of cells (particles). We developed a new approach to simultaneously compute field values such as turgor pressure, stress and strain rates while predicting the establishment of the root anatomy during growth. We show the influence of soil mechanical constraints on the development of the root and the reorganisation of the cell architecture in the meristem. The model could also predict growth patterns observed experimentally and use 3D microscopy data as input for numerical simulations.
In the future, our model will foster a better understanding of complex soil biological processes that are especially difficult to observe. This technique has the potential to incorporate the key biophysical processes of the root in a robust root-soil interaction model.
CT06 Porous media
Giorgos Minas (St Andrews)
Stochastic modelling, simulation and analysis of oscillatory biological systems: the NF-κB case study
Cells constantly receive a multitude of different signals from their external environment. They use networks of interacting molecules to respond to these signals and trigger the appropriate actions. An important target of molecular biology is to identify and study the key components of these networks that are often found to be therapeutic targets. An important example is the NF-κB signalling system that responds to a variety of signals related to stress and inflammation in order to activate a large number (500) of different genes. The NF-κB network is noisy and complex with oscillatory dynamics, involving multiple feedback loops, and therefore, mathematically, very interesting. In this talk, I am going to introduce the NF-kB signalling pathway, discuss a stochastic model for oscillatory systems and describe an analytical framework for assessing the ability of a stochastic signaling system to distinguish between simultaneously received signals.
CT 16 Mathematical Biology-3
Andrew James Mitchell (Strathclyde)
The effect of the Lower Boundary on Porous Squeeze-Film Flow
Squeeze-film flow of a Newtonian fluid (that is, flow of a layer of fluid in the gap between two rigid impermeable plates that approach each other) is a classical problem in fluid mechanics, with applications in, for example, the squeezing of synovial fluid in the knee joint or between hydrolic clutch plates. As is well known, an infinite time is required for all of the fluid to be squeezed out of the gap, i.e. for the plates to make contact. Knox et al. (“Porous Squeeze-Film Flow”, IMA J. Appl. Math. 80 2015, 376-409) generalised classical squeeze-film flow to the case of fluid being squeezed between an impermeable plate and a porous layer with an impermeable base. In particular, using the Darcy equation in the porous layer and the Navier–Stokes equation in the fluid layer, and imposing the Beavers–Joseph condition at the interface between the fluid and porous layer, Knox et al. (2015) showed that the contact time is finite and depends on the permeability of the porous layer. The current work extends that of Knox et al. (2015) for both an axisymmetric and a two-dimensional geometry to consider
what happens when there is no impermeable base below the porous layer. Like Knox et al. (2015), we find that there is a finite contact time, which depends on the permeability of the porous layer. The behaviour in the limits of small and large permeability is also analysed. We find that as a result of the lower impermeable base being removed the contact of the impermeable plate and the porous layer happens much sooner than in the problem considered by Knox et al. (2015).
CT06 Porous media
Sunil Modhara (Nottingham)
Neural fields with rebound currents: novel routes to patterning
The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. To replicate this success for sub-cortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies, and most importantly the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one- and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed in the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from non-smooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic travelling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic travelling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution.
Poster
Joshua Moore (Cardiff)
Morphology driven bifurcations: graphical approaches to pattern formation via lateral feedback dynamics in bilayer geometries
Over the past two decades, fine-grain patterns produced by juxtacrine signalling have been studiedusing static monolayers as cellular domains, where analytical results are limited to a few cells due to the algebraic complexity of the required nonlinear dynamical systems. Motivated by concentric patterning of Notch expression observed in the mammary gland, we discuss how to represent the global dynamical system as a network of contrasting subsystems in order to exploit the symmetric structure of the observed patterns in both 2D and 3D. We then extend the existing theory of static monolayer pattern formation to derive analytical conditions for the existence and stability of laminar patterns in the bilayer. Critically, we show that these results are independent of the precise lateral-feedback model and therefore provide a powerful tool to investigate the influence of tissue geometry on patterning capacity using lateral-feedback mechanisms. Applying the analytic conditions to mammary tissue structures suggests that intense cell signalling polarity is required for the maintenance of stratified cell-types within a static bilayer using a lateral-inhibition mechanism. Furthermore, by employing 2D and 3D cell-based models, we highlight that the cellular polarity conditions derived from static domains have the capacity to generate laminar patterning in dynamic environments. However, they are insufficient for the maintenance of patterning when subjected to substantial morphological perturbations. Finally, we discuss a natural extension of the signal pattern existence and stability results to include asymmetric connectivity, which allows us to analyse the impact of bilayer curvature on morphological symmetry breaking events.
MS29 IMA Lighthill Thwaites
Matthew Moore (Oxford)
How solute diffusion counters advection in the early stages of coffee ring formation
We study the initial evolution of the coffee ring that is formed by the evaporation of a thin surface tension-dominated droplet containing a dilute solute. When the solutal Péclet number is large, we show that diffusion close to the droplet contact line controls the coffee-ring structure in the initial stages of evaporation. We perform a systematic matched asymptotic analysis for two evaporation models - a simple, non-equilibrium, one-sided model (in which the evaporative flux is taken to be constant across the droplet surface) and a vapour-diffusion limited model (in which the evaporative flux is singular at the contact line) - valid during the early stages in which the solute remains dilute. We call this the 'nascent coffee ring' and describe the evolution of its features, including the size and location of the peak concentration and a measure of the width of the ring. We consider a variety of droplet shapes to asymptotically investigate the role played by contact line curvature in the formation of the ring. Moreover, we use the asymptotic results to investigate when the assumption of a dilute solute breaks down and the effects of finite particle size and jamming are expected to become important.
CT14 Droplets
Piotr Morawiecki (Bath)
Asymptotic framework for flood models comparison
Many state-of-the-art approaches for estimating flood risk includes physical, conceptual, and statistical modelling. Despite their overall good performance, it is observed that data-based modelling approaches at some situations give inaccurate predictions, especially in conditions underrepresented in the training data. Understanding the limits of these models applicability remains an open challenge. Here we present a unified framework using asymptotic analysis, which highlights differences between these modelling approaches. It provides clear analytical and numerical benchmarks on the different approaches in a range of scenarios. In consequence, the proposed approach may lead to a better understanding of uncertainties in hydrologic models, and development of more theoretically-justified flood estimation methods.
Poster
Antonio Moro (Northumbria)
Dispersive shock states in matrix models
We show that Hermitian Matrix Models with arbitrary degree of nonlinearity support the occurrence of a phase transition described by a dispersive shock solution of a nonlinear dispersive hydrodynamic system. The order parameters, defined as derivatives of the free energy with respect to the coupling constants can be obtained as a solution of the Toda lattice equations.
The thermodynamic limit corresponds to the continuum limit of the Volterra system where the order parameter is a solution of a nonlinear partial differential equation in a small dispersion regime. The order parameter evolves in the space of coupling constants as a nonlinear wave that develops a dispersive shock for a suitable choice of the couplings. Our analysis explains the origin and the mechanisms leading to the emergence of chaotic behaviours observed by Jurkiewicz in M^6 matrix models.
MS23 Dispersive hydrodynamics and applications
Nigel Mottram (Glasgow)
Current research in liquid crystalline materials
In this talk I will introduce the main areas of current research in liquid crystal theory and the application of this theory to explain liquid crystal phenomena. In particular, I will attempt to place the talks contained in the minisymposium in the context of past and ongoing research in this area – summarising modelling approaches from the molecular to the macroscopic length scales and focussing on the theoretical frameworks that allow accurate modelling of self-organisation, defect formation and movement, and active fluids. I will also summarise ideas for the further development of these theories, including holes in our current understanding of liquid crystal, and related, systems.
MS22 Theory and modelling of liquid crystalline fluids
Anthony Mulholland (Bristol)
Ultrasonic Waves in Stochastic Layered Elastic Media
This talk considers the propagation of high frequency elastic waves propagating the time domain in a layered polycrystalline material. This work is important in deepening our understanding of ultrasonic non-destructive testing of composites and polycrystalline components such as those found in the nuclear and aerospace industries. Importantly we will study a regime where the heterogeneous media is such that significant multiple scattering occurs and the transmitted energy is primarily contained in the incoherent component of the transmitted wave. This extended coda wave (and lack of a meaningful coherent component) mean that a homogenisation approach cannot be applied. Instead a probabilistic approach is adopted whereby the moments of the amplitude of the transmitted wave can be studied. Each layer is locally anisotropic and the layer thicknesses and crystal orientations follow a stochastic (Markovian) process. We consider the propagation of monochromatic shear waves and the resulting stochastic differential equations lead to a self-adjoint infinitesimal generator and Fokker-Planck equation via a limit theorem approach. Explicit expressions for the moments of the probability distributions of the power transmission and reflection coefficients are then derived. Modern ultrasonic non-destructive testing employs an array of transmitting and receiving elements so that the beam can be steered. In this talk we will therefore examine the dependency of the ultrasonic wave propagation on the wavevector direction.
MS10 Ultrasonic Waves
Shibabrat Naik (Bristol)
Tilting and Squeezing: Implications of Hamiltonian saddle-node bifurcation for reaction dynamics
Authors: Shibabrat Naik, Víctor J. García-Garrido, Wenyang Lyu, Stephen Wiggins
Hamiltonian models provide insights into phase space mechanisms of chemical reactions which have a characteristic potential energy surface. We present the normal form of a two-degree-of-freedom Hamiltonian that undergoes saddle-node bifurcation. We discuss how the changes in the geometry, that is depth and flatness, of the potential energy surface leads to the saddle-node bifurcation. Then, we show how the bifurcation affects the geometry of the phase space structures that mediate reactions in this Hamiltonian system. We discuss how Lagrangian descriptors can be used for detecting the qualitative changes in the phase space structures with changes in the depth and flatness of the potential energy surface. Then, we address the qualitative and quantitative changes due to depth and flatness from a reaction dynamics perspective.
CT13 Dynamical Systems
Michael Negus (Oxford)
High-speed droplet impact onto deformable substrates: analysis and simulations
The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.
CT14 Droplets
Matthew Nethercote (Manchester)
Edge Diffraction of Acoustic Waves by Periodic Composite Metamaterials: The Hollow Wedge
Normally in metamaterial research, some form of infinite periodicity is assumed which allows us to restrict the study to a small portion known as the unit cell. This has led to many studies which increase the complexity of the unit cell and reconstruct the global scattering using the periodicity of the metamaterial. An alternative approach looks into the case where infinite periodicity is no longer assumed. This means that the metamaterial will have well-defined boundaries that can symbolise many different interfaces such as edges and corners.
In this presentation, the scattering of an acoustic pressure wave by a hollowed out wedge is studied where, for simplicity, the unit cells will be sound-soft cylinders with infinite height and a small radius. This configuration can also be viewed as two separate semi-infinite gratings with two sets of scattering coefficients to determine. We will construct an iterative scheme from the resulting infinite system of equations and find a solution using the discrete Wiener-Hopf technique. We shall also discuss some tools that are useful for computations such as tail-end asymptotics and rational approximations.
CT12 Waves
Kathryn Nightingale (Duke)
3D Shearwave Elasticity Imaging
Elasticity imaging methods mechanically perturb tissue, image the dynamic tissue response, and reconstruct images of the underlying tissue mechanical properties using material models of varying complexity. One such method, shearwave elasticity imaging (SWE), employs focused acoustic beams to generate an acoustic radiation force impulse (ARFI) to locally push on tissue and uses the same diagnostic ultrasound transducer to monitor the resulting shearwave propagation, typically along a single imaging plane. A major focus of our laboratory has been the development and implementation of high resolution 3D ARFI/SWE elasticity imaging methods. One application is in prostate cancer imaging and targeted biopsy guidance, where our initial in vivo findings have demonstrated that ARFI imaging is specific for clinically significant prostate cancer, and we are developing a dedicated transrectal ultrasonic ARFI/SWE biopsy guidance system. In another application, we are developing 3D SWE tools employing higher order material models and advanced data acquisition and processing methods in order to better characterize anisotropic and dispersive materials. We have obtained preliminary 3D SWE data in human vastus lateralis muscle in vivo in which we have observed the propagation of multiple shearwave modes. These modes can both confound shearwave speed estimation during planar imaging as well as providing the opportunity to more fully characterize the anisotropic properties of muscle. In this talk I will summarize our current finding, and discuss future clinical opportunities.
MS10 Ultrasonic Waves
Anthony Nixon (Lancaster)
Graph rigidity and flexible circuits
A framework is a geometric realisation of a graph in Euclidean d-space. Edges of the graph correspond to bars of the framework and vertices correspond to joints with full rotational freedom. The framework is rigid if every edge-length preserving continuous deformation of the vertices arises from isometries of d-space. Generically, rigidity is a rank condition on an associated rigidity matrix and hence is a property of the graph which can be described by the corresponding row matroid. Characterising which graphs are rigid is solved in dimension 1 and 2, but open in dimension at least 3. One fundamental problem in higher dimensions is the existence of flexible circuits; these are non-rigid graphs that are circuits in this matroid.
I will give a survey of graph rigidity including recent joint work with Georg Grasegger, Hakan Guler and Bill Jackson where we analyse flexible circuits in dimension d.
BMC06 Combinatorics
Andrew Norris (Rutgers)
Stress based elastic waves
Elastic waves are usually considered as travelling displacement perturbations. But they are also propagating stress waves. In this talk we look at the implications of viewing elastic wave motion in terms of stress as the fundamental variable. Several results will be discussed: (1) stress wave solutions, (2) stress based acoustoelasticity, and (3) implications for nonlinear elasticity of cubic symmetry.
First, (1), we look at how the stress formulation yields three propagating solutions, in agreement with the standard displacement formulation. Since stress is six-dimensional, this implies three solutions with zero wave speed, which we call non-propagating waves. We show how the non-propagating solutions fall out of a six-dimensional formulation of the dynamic problem.
(2) Acoustoleasticity, which considers the effect of pre-stress on wave motion, is quite naturallydescribed in the stress formulation. Of the six stress waves solutions from (1), the three propagating ones satisfy the expected shift in wave speed as functions of the pre-stress. In contrast, the three non-propagating solutions are perturbed from zero velocity into waves with speeds proportional to the square root of the pre-stress. This square root can be real or imaginary, implying slow propagating purely acoustoelastic waves, or locally exponentially growing solutions.
Finally, (3) we discuss implications for cubic elasticity. The stress viewpoint suggests an optimal way to define the third order elasticity in terms of hydrostatic and generalized deviatoric components.
MS10 Ultrasonic Waves
Reuben O'Dea (Nottingham)
Aiding design of 3D printed bioactive tissue engineering scaffolds via multiscale and multiphase modelling
The fabrication of tissue engineering scaffolds for bone implants via additive manufacturing technologies holds great promise for accelerating the clinical translation of in vitro tissue engineering approaches for bone regenerative therapies. Such approaches provide detailed control over micro- and macro-scopic scaffold architecture, and the ability to functionalise with growth factors such as BMP2 that promote differentiation of seeded stem cell populations down osteogenic pathways. However, high cost and side effects limits the amount of BMP2 that may be used. To seek to optimise scaffold pore design, we first combine a simple computational model of pore-scale tissue growth with a description of its macroscopic mechanics. Subsequently, we employ a multiphase modelling approach (informed by multiscale homogenisation techniques) and tailored in vitro experiments, to understand how scaffold pore design, nutrient transport and distributions of BMP2 may be tailored to promote osteogenesis.
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Casper Oelen (Loughborough)
Automorphic Lie algebras on tori
An automorphic Lie algebra (aLia) is a Lie algebra of certain invariants which arises in the theory of integrable systems. More specifically, aLias were introduced in the context of reduction of Lax pairs. ALias are defined as follows. Let a finite group G act on a compact Riemann surface and on a complex finite dimensional Lie algebra, both by automorphisms. Consider the space of meromorphic maps from the surface to the Lie algebra with poles restricted to a finite set. The subspace of G-equivariant maps is an automorphic Lie algebra.
They are a generalisation of twisted loop algebras and arise in seemingly unrelated contexts throughout mathematics. Although the motivation of studying these algebras originated from the theory of integrable systems, the theory of aLias has now become a subject of its own interest.
To obtain an understanding of an aLia, one would like to find a certain normal form. Once this is found, questions about isomorphisms between aLias are considerably easier to answer. One of the long standing conjectures of this subject is the question about the existence of such a normal form. For the Riemann sphere, considerable effort has been made over the past decades and the existence of a normal form has been found under certain assumptions. The case of other Riemann surfaces such as complex tori, has not been fully explored yet and is one of the aims of my research.
This research, which is joint work with Sara Lombardo and Vincent Knibbeler, is focused on aLias based on Riemann surfaces of genus 1, that is, complex tori. The finite subgroups of the automorphism group of the torus are given by (semi) direct products of certain cyclic groups. Not much is known yet about this class of aLias and my current research focuses on gaining an understanding of these types of infinite dimensional Lie algebras, in particular where the groups G are the dihedral groups.
In this research, we present the first example of an aLia where G is non-abelian and give a concrete description of the algebra.
Poster
Koji Ohkitani (Sheffield)
Revisiting self-similarity for the 3D Navier-Stokes equations
We investigate the scale-invariance property the Navier-Stokes equations theoretically and numerically, as a basis for studying their statistical solutions. (See a related reference below.) We distinguish critical scale-invariance of 'the first kind' (deterministic) and that of 'the second kind' (statistical), the latter of which is related to the so-called source-type solution (that is, a nonlinear counterpart of fundamental solutions to linear PDEs.)
Mathematically a self-similar solution is known to exist in a function class which contains singular solutions like 1/|x|, but its explicit functional form is yet to be determined. We formulate a forward self-similar problem for the 3D Navier-Stokes equations in velocity, vorticity and vorticity gradient and introduce successive approximations. We work out the self-similar decaying profile to the leading-order and determine it approximately to compare it with those of the Burgers and 2D Navier-Stokes equations.
(This is a joint work with Dr Riccardo Vanon.)
Reference:
K. Ohkitani, "Study of the Hopf functional equation for turbulence:
Duhamel principle and dynamical scaling," Phys. Rev. E 101, 013104(2020).
CT07 High Reynolds Number
Katie Oldfield (National Museums Scotland)
Maths Week Scotland
Maths Week Scotland is a week of events and activity, with special events throughout the year. Now in its fifth successful year, Maths Week Scotland will take place 27 September - 3 October 2021.
Maths Week Scotland is for everyone! We believe in creating opportunities to experience maths as relatable, exciting and relevant. Events take place all across Scotland for families, adults and schools during Maths Week Scotland.
See our highlights of Maths Week Scotland 2020 in this blog post www.mathsweek.scot/news/our-maths-week-scotland-2020-highlights including maths in nature, Olympic athletes, competitions, and puppies wearing badges.
There are plenty of ways you can get involved with Maths Week Scotland. First way is to contact the engagement representative at your department or university. We work with most universities in Scotland already and they will know of ways to get involved not just during the week itself but year round. If you aren’t sure who that is then contact info@mathsweek.scot This year we also have some training available for staff, post graduate students and third year and above undergraduate students. Again contact info@mathsweek.scot to find out more. Closing date for applications is 8 April. You can find out more about Maths Week Scotland at www.mathsweek.scot or follow us on Twitter [@MathsWeekScot](https://twitter.com/MathsWeekScot)
Outreach Video
Mette Olufsen (North Carolina)
Uncertainty in Patient-Specific Network Models for Pulmonary Hypertension
Cardiovascular disease management involves interpretation of imaging data, time-series data, and single-valued markers often measured over several visits. While each data type provides insight into the disease state, these snapshots cannot easily be integrated to provide insight into disease predictions. In this study, we demonstrate how to predict disease state using a fluid mechanics model that integrates computed tomography (CT) images with blood pressure measurements from right heart categorization. To demonstrate our methodology, we fit the model to data from control and hypertensive mice exposed to a prolonged period of hypoxia. We use this model to characterize patient-specific remodeling in the proximal and distal vasculature. The large vessels are represented explicitly by their length, radius, and connectivity, while structured trees represent the small vessels. We calculate nominal values for vessel stiffness and hemodynamic parameters from morphometric and invasively measured hemodynamic data from control and hypertensive mice, and we use a Bayesian approach to estimate subject-specific parameters fitting the model to data. This type of parameter estimation allows us to propagate the uncertainty of pressure and flow predictions to all large vessels. In addition to time-series predictions of pressure and flow, we validate our results in the frequency domain assessing change in wave-propagation and wave-intensity with disease. For the micro-vasculature, we conduct a morphometric analysis characterizing changes in branching structure of the arterial networks. This is done by extracting skeletonized networks from the micro-CT images and using a custom algorithm to represent the network as a connected graph. We determine subject-specific fractal parameters and analyze how these changes with PH. Our model and data analysis outcomes are combined to understand the link between spatially distributed etiologies and global hemodynamics and shed light on the prospect of combining the model and graph-based morphometric analysis of vascular trees.
MS27 Mathematical and Computational Modelling of Blood Flow
Ifeanyi Onah (Glasgow)
Predicting the onset of retinal haemorrhage
Retinal haemorrhage occurs due to abnormal bleeding of the blood vessels in the retina, often triggered by injury in the brain. We use a theoretical model to test a clinical hypothesis for the onset retinal haemorrhage, where bleeding results from a rise in intracranial pressure (ICP) in the brain. This pressure rise is transmitted into the eye via the optic nerve sheath, leading to a proportionate increase in the venous pressure, which then spreads through the retinal vasculature and leads to vessel bursting. In this poster we develop a mathematical model for flow in a single blood vessel with discontinuous material properties, to mimic the course of the central retinal vein as it enters the eye. We formulate a nonlinear Riemann solver (based on Newton's method) to describe the evolution of a step disturbance initiated at the point of discontinuity in vessel properties. We quantify the structure of the flow and the resulting wall profiles for cases where the flow speed is less than the wavespeed (subcritical).
Poster
Timothy Ostler (Cardiff)
Computational Modelling of temperature during vitrification of multiple oocytes/embryos
Vitrification, or ultrarapid freezing, is a well-studied method used for cryopreservation of gametes and embryos Within an IVF laboratory, it is becoming common practice to vitrify multiple oocytes (and sometimes embryos) on a single vitrification device. However, there is currently little guidance on how this practice affects cooling rates of oocytes, or embryos, often used as a predictor of survival rates.
Poster
Michela Ottobre (Heriot-Watt)
Uniform in time approximations of Stochastic Dynamics
Complicated models, for which a detailed analysis is too far out of reach, are routinely approximated via a variety of procedures, for example by use of numerical schemes. When using a numerical scheme we make an error which is small over small time-intervals but it typically compounds over longer time-horizons. Hence, in general, the approximation error grows in time so that the results of our simulations are less reliable when the simulation is run for longer. However this is not necessarily the case and one may be able to find dynamics and corresponding approximation procedures for which the error remains bounded, uniformly in time. We will discuss some criteria and approaches to understand when this is possible. We will start by considering the simple case of approximations produced via the Euler scheme and then, time allowing, consider more general approximation procedures, i.e. averaging.
MS02 Mathematics for Data Science
Markus Owen (Nottingham)
Modelling and Analytics for a Sustainable Society - Food, Cities and Water
Society faces an ongoing global problem of food shortages, water scarcity and insufficient clean energy, exacerbated by climate change. These resource challenges are interconnected, highly complex and nonlinear, and inseparable from their social context. Mathematics has an underexploited role to play in addressing these issues by quantifying and predicting the effects of alternative approaches and interventions. In this talk I will give an overview of the importance of mathematics for sustainability and discuss diverse specific examples including the importance of crop root systems for food security, spatial structure and inequality in cities and the impact of solar geoengineering on water scarcity.
MS20 Mathematics of the water, energy and food security nexus
Sergio P. Perez (Imperial)
Unconditional bound-preserving and energy-dissipating finite-volume schemes for the Cahn-Hilliard equation
The Cahn-Hilliard (CH) equation is a popular phase-field model initially proposed by Cahn and Hilliard to mathematically simulate the process of phase separation in binary alloys. Since then it has found applicability in a myriad of modelling problems as diverse as image inpainting, diblock copolymer molecules, capillarity and wetting phenomena, tumor growth, topology optimization and many more. We propose finite-volume schemes for the CH equation that unconditionally and discretely satisfy the boundedness of the phase field and the free-energy dissipation. Our scheme is applicable to a variety of free-energy potentials including the Ginzburg-Landau and Flory-Huggins, general wetting boundary conditions accounting for wall free energies, and degenerate mobilities. The thrust of our scheme relies on its finite-volume upwind methodology, which we combine with a semi-implicit formulation based on the classical convex-splitting approach for the free-energy terms. Moreover, it can be conveniently extended to an arbitrary number of dimensions thanks to its cost-saving dimensional-splitting nature, which allows to efficiently solve higher-dimensional simulations with a straightforward parallelization. The scheme is validated and tested out in a variety of prototypical configurations with different number of dimensions, where the inclusion of wall free energies leads to a rich variety of contact angles between droplets and substrates. This is a joint work with Rafael Bailo, José A. Carrillo and Serafim Kalliadasis.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Martin Palmer-Anghel (Bucharest)
On homological stability for configuration-section spaces
Configuration-mapping spaces, introduced by Ellenberg, Venkatesh and Westerland, are spaces of configurations of points on a manifold, together with a continuous map from the complement of the configuration to a fixed space, with prescribed "monodromy" in a neighbourhood of the configuration points. This naturally generalises to configuration-section spaces, where the complement is equipped with a section of a given bundle over the manifold, such as a non-vanishing vector field. These spaces may be interpreted physically as spaces of "fields" in an ambient manifold, which are permitted to be singular at a finite number of points, with prescribed behaviour near the singularities.
Mathematically, the first such spaces to be considered were Hurwitz spaces, which may be viewed as certain configuration-mapping spaces on the 2-disc. Ellenberg, Venkatesh and Westerland proved that, under certain specific conditions, Hurwitz spaces are (rationally) homologically stable; from this they were able to deduce an asymptotic version of the Cohen-Lenstra conjecture for function fields, a purely number-theoretical result.
We will present a higher-dimensional analogue of their stability result, namely: homological stability (with integral coefficients) for configuration-section spaces on an ambient manifold M of dimension at least 3, as long as M is either simply-connected or its geometric dimension and its handle dimension differ by at least 2.
This represents joint work with Ulrike Tillmann.
Poster
Jasmina Panovska-Griffiths (UCL/Oxford)
Application of mathematical modelling to COVID-19 epidemic waves in the UK
Since the onset of the COVID-19 epidemic, I have been leading modelling work related to evaluating different strategies to control the spread of SARS-CoV-2 and its variants in the UK. My talk will describe some of this work focusing on an application of an agent-based model called Covasim to the UK epidemic during 2020-2021. Firstly, I will showcase modelling work describing the impact of the effectiveness of masks and different test-trace strategies during the first COVID-19 wave and in absence of virus variants. Secondly, I will illustrate the application of Covasim to quantify the transmissibility of the B.1.17 variant during the second epidemic wave and the need for a second national lockdown. Finally, I will illustrate the application of Covasim to evaluate whether the current vaccination strategy will be sufficient to control COVID-19 resurgence in late 2021 under the planned roadmap to reopening from March 08, 2020.
MS28 Covid-19 Modelling
Emilian Parau (East Anglia)
Hydraulic falls and trapped waves over topography
Steady two-dimensional free-surface flows past submerged obstructions on the bottom of a channel are presented. The fluids is either homogenous or consists of two layers with different densities. Both the effects of gravity and surface tension are considered. Flexural-gravity hydraulic falls are also found when the fluid is covered above by a thin elastic plate. The evolution in time of the hydraulic falls is analysed using a time-dependent numerical algorithm.
Co-author: Charlotte Page
MS14 Variational Methods in Geophysical Fluid Dynamics
William Parnell (Manchester)
Elastostatic cloaking, low frequency elastic wave transparency and neutral inclusions
Cloaking has been of interest for decades although only recently have advances been made in achieving this in specific physical scenarios, including acoustics, electrodynamics and elastodynamics. The latter is more difficult than the former two areas and it transpires that a new theoretical framework is required for invariance of the governing equations. In principle, an elastodynamic metamaterial cloak should work at any frequency and for any wave type and it should conceal any object interior to it thus being independent of its nature and geometry. Practically speaking however it is impossible to achieve such properties. Here we take a step back and reconsider the theory of elastostatic cloaking and the limit of low-frequency transparency (LFT). We tie these concepts to the neutral inclusion (NI): a coated inclusion with coating designed to render the inclusion invisible to a specific loading. The dependence of coating on inclusion properties is what makes the NI different to ideal (currently unachievable) metamaterial designs, although NIs have the benefit of being more practical.
NIs are understood for hydrostatic loadings but this is not the case for shear-type loadings. Although “imperfect boundary conditions” (difficult to achieve) can yield neutrality, it has been thought, even until recently that finite thickness coatings cannot ensure neutrality. Here we address this problem, employing the impedance matrix approach to the two-dimensional equations of elastostatics and obtaining conditions on coating properties so that NIs act neutrally for both hydrostatic and planar shear loading. The NI coating is found to require anisotropic properties. We provide links to cloaking and LFT, noting in particular (and perhaps non-intuitively) that leading order LFT is not equivalent to an elastostatic cloak in general. We thus introduce the concepts of weak and strong NIs and show that the well-known generalised self-consistent method can be considered equivalent to weak NIs.
MS10 Ultrasonic Waves
Robert Patterson (Weierstrass)
Decomposing large deviations rate functions into reversible and irreversible parts
We study the large deviations of mean field interacting particle systems without any assumptions of reversibility or detailed balance.
We show that the resulting rate functions admit interesting decompositions into reversible and irreversible parts. As a consequence we obtain some entropy/free energy inequalities generalising the role of the Fisher Information in more classical problems. Our decomposition also suggests a way of representing the limiting dynamics of the particle systems in terms of a reversible process plus an irreversible process.
Joint work with Michiel Renger and Upanshu Sharma
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Philip Pearce (Harvard Medical School)
Emergent robustness of bacterial quorum sensing in fluid flow
Bacteria use intercellular signalling, or quorum sensing (QS), to share information and respond collectively to aspects of their surroundings. The autoinducers that carry this information are exposed to the external environment; consequently, they are affected by factors such as removal through fluid flow, a ubiquitous feature of bacterial habitats ranging from the gut and lungs to lakes and oceans. Here, we develop and apply a general theory that identifies and quantifies the conditions required for QS activation in fluid flow by systematically linking cell- and population-level genetic and physical processes. We predict that, when a subset of the population meets these conditions, cell-level positive feedback promotes a robust collective response by overcoming flow-induced autoinducer concentration gradients. By accounting for a dynamic flow in our theory, we predict that positive feedback in cells acts as a low-pass filter at the population level in oscillatory flow, allowing a population to respond only to changes in flow that occur over slow enough timescales. Our theory is readily extendable, and provides a framework for assessing the functional roles of diverse QS network architectures in realistic flow conditions.
CT10 Mathematical Biology-2
Nicholas Pearce (Coventry)
Modelling cardiac mechano-electric feedback response to perturbations using a modified multi-scale model of heart dynamics.
Cardiac dynamics are a function of many scales and physical processes housed entirely within the heart itself. Feedback mechanisms across these scales helps maintain the regulation of blood flow. The two most important mechanisms are the mechanoelectric feedback (MEF) and the electro-contraction coupling (ECC). In the former, microscopic changes in the mechanical environment of the muscle cell starts electrochemical processes leading to alterations in the macroscale mechanical regulation of blood flow. The ECC can be described as the opposite to the MEF. Whilst MEF helps maintain heart stability, it has been shown to both promote and discourage arrhythmia and asystole. In some instances, a mechanical stimulation of the heart may even lead to death.
Here, a multiscale lumped-parameter model of the heart that incorporates the MEF mechanism is modified to investigate the response to a sudden mechanical stimulation of muscle cells. The model has previously shown chaotic behaviour resulting from a dysfunction of MEF. We find that that the timing of an impulse is a critical indicator for the resulting behaviour. Though the predominant response to an impulse is found to be short term arrhythmia, it is also found that a short mechanical perturbation delivered during diastole can regularise a dysfunctional heart. This contrasts with recent studies finding mechanical stimulation leads only to fibrillation.
CT 16 Mathematical Biology-3
Lorenzo Pellis (Manchester)
Challenges in short-term projections of COVID-19 dynamics to inform the UK pandemic response
I will present a model of COVID-19 dynamics in the UK that is used for the weekly estimates of Rt and short-term projections that, once combined with other models’ estimates, are published by SAGE. I will discuss the most interesting and novel aspects of the model structure and how this is parameterised from within-hospital line-list data and nation-wide data about hospital admission and bed occupancy. I will focus in particular on practical data issues and model limitations. Time permitting, I will illustrate how similar ideas were instrumental to raising the alarm bell that likely anticipated the UK lockdown by a few days, avoiding the NHS being overwhelmed.
MS05 Multiscale Modelling of Infectious Diseases
Sarah Penington (Bath)
Genealogies in pushed waves
Consider a population in which each individual carries two copies of each gene, and suppose that a particular gene occurs in two different types, a and A. Suppose that individuals carrying AA have a higher evolutionary fitness than aa individuals, and that aA individuals have a lower evolutionary fitness. We can model this situation using a stepping stone model on the integers, and show that (under certain conditions) as the number of individuals at each site goes to infinity, the genealogy of a sample of type A genes from the population (under a suitable time scaling) converges to a Kingman coalescent.
Joint work with Alison Etheridge.
MS12 Front Propagation in PDE, probability and applications
Daniel Peralta-Salas (ICMAT)
Helicity uniqueness in 3D Hydrodynamics and MHD
Helicity is a remarkable conserved quantity that is fundamental to all the natural phenomena described by a vector field whose evolution is given by a volume-preserving flow. This is the case of the vorticity of an inviscid fluid flow or of the magnetic field of a conducting plasma. In this talk I will report about recent work proving that helicity is the only regular integral invariant of volume-preserving diffeomorphisms. I will also show that an analogous result holds for the pair formed by magnetic helicity and cross helicity (under the action of the MHD group). This is based on joint works with A. Enciso, B. Khesin, F. Torres de Lizaur and C. Yang.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
Cédric Pilatte (ENS (Paris))
The inverse slope problem for sets in general position and additive combinatorics
A famous problem in additive combinatorics is to find the smallest possible size of the sumset A+A, where A is a subset of n elements an abelian group G. The corresponding inverse problem is to classify the sets A of size n where |A+A| is minimal (or close to the minimal value). Several well-known theorems state that, under suitable assumptions on G and n, A must be (a large subset of) an arithmetic progression.
In discrete geometry, the slope problem is the question of finding the least number of distinct slopes that are determined by a set of n points, not all on the same line. The classification of configurations of points which achieve or are close to this minimum will be referred to as the inverse slope problem.
We will consider the inverse slope problem when the condition "not all collinear" is replaced by "in general position". In this case, surprising connections arise between the inverse slope problem and the inverse problem in additive combinatorics. The solutions to the inverse slope problem are (large subsets of) the vertex sets of affinely regular polygons. Any non-degenerate conic can be given the structure of an abelian group, and these configurations correspond to arithmetic progressions on ellipses.
Poster
Aaron Pim (Bath)
The prediction of defect strengths in nematic shells
Nematic liquid crystals are an intermediate phase of matter between crystalline solid and isotropic liquid states, whose molecules closely resemble rods [11, 4]. A nematic shell is a thin fillm of nematic liquid crystal coating a curved substrate, which is a modelled as a smooth two-dimensional manifold which is embedded into three-dimensions [10]. Liquid crystals can also possess defects in their patterns, localised discontinuities which can occur as either isolated points or as lines, also known as disclination lines, [2].
A natural way to model the preferred direction of orientation would be a vector field with a corresponding energy functional, in the same manner toOseen-Frank [5]. We can consider a simplified form of this energy functional known as the surface one-constant approximation, which is a reasonable approximation in certain cases [6, 9]. In this model the presence of defects results in a divergence of the energy density, thus we regularise by excluding smallregions about each of the defects.
There are topological laws that vector fields, and by extension our liquid crystal, must obey. One of which is the Poincare{Hopf theorem, which relates the Euler characteristic of the surface to total strength of the defects on the surface [1, 3]. However, this theorem is only applicable to closed manifolds, ones that do not possess a boundary. I shall present a generalisation of the Poincare{Hopf theorem, which accounts for manifolds with boundaries and boundary defects. Then I shall utilise the divergence of the energy density about defects, to predict the expected strengths of a system given only the geometerical parameters. This shall answer the question as to why there are multiple stable configurations of defects in shallow wells [8, 7] and give a computational method of prediction for other well shapes.
References
[1] Manfredo P.do Carmo. Differential Geometry of Curves and Surfaces.Prentice-Hall Inc., 1976.
[2] S. Chandrasekhar. Liquid Crystals. Cambridge University Press, 2nd edition, 1992. 1
[3] Diarmuid Crowley and Mark Grant. The poincare{hopf theorem for line fields revisited. Journal of Geometry and Physics, 117:187{196, 7 2017.
[4] P.G. de Gennes. The physics of Liquid Crystals. The international seriesof monographs of Physics. Clarendon Press, 1974.
[5] F.C.Frank. I. liquid crystals. on the theory of liquid crystals. Discussionsof the Faraday Society, 25:19{25, 1958.
[6] F. Leenhouts and A.J.Dekker. Elastic constants of nematic liquid crystalline schiff's bases. The Journal of Chemical Physics, 74, 1981.
[7] Alexander Lewis. Defects in Liquid Crystals: Mathematical and Experimental Studies. PhD thesis, St Anne's College University of Oxford, 2015.
[8] Chong Luo, Apala Majumdar, and Radek Erban. Multistability in planar liquid crystal wells. Physical review, 85, 2012.
[9] M. More, C. Gors, P. Derollez, and J. Matavar. Crystal structure of 4-methoxybenzylidene-4-n-butylaniline mbba the c4 and c3 phases. Liquid Crystals, 18:337{345, 1995.
[10] Gaetano Napoli and Luigi Vergori. Surface free energies for nematic shells.Physical Review, 85, 2012.
[11] E.G. Virga. Variational Theories for Liquid Crystals, volume 8 of AppliedMathematics and Mathematical Computation. Chapman and Hall, 1st edition, 5 1995. 2
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Inna Polichtchouk (ECMWF)
Spontaneous inertia-gravity wave emission from a nonlinear critical layer in the stratosphere
Inertia gravity waves (IGWs) transport heat and momentum and contribute to mixing in the atmosphere. Spontaneous emission of IGWs by a flow that is initially well balanced provides an important source of nonorographic gravity waves in the atmosphere. This study investigates spontaneous IGW emission in an idealized representation of the winter stratosphere using a nonlinear global model. It is shown that IGWs are spontaneously emitted in a polar night jet exit region that develops around a nonlinear Rossby wave critical layer, where planetary scale Rossby waves break. The key ingredients for IGW generation are identified and the evolution of IGWs following the emission is discussed. Part of the emitted IGWs remain captured in the Rossby wave critical layer, but another part — in a form of a well-defined IGW packet — propagates away into the far field. The propagating wave packet is numerically well-converged to increases in both vertical and horizontal resolution. Thus this setup provides an ideal test bed for understanding IGW emission and informing design of nonorographic gravity wave drag parametrizations included in many numerical weather and climate prediction models which do not explicitly resolve IGWs.
MS16 Eddies in Geophysical Fluid Dynamics
Satya Prakash Pradhan (Hyderabad, India)
Buckling of chiral rods due to coupled axial and rotational growth
Most existing works modelling nonplanar configurations in growing filaments stick to isotropic rods. Such models usually rely on differential growth, or the presence of an external elastomeric matrix, multirod composites, or phototropism to model the generation of curvature and torsion in nonplanar deformations. Growth in chiral rods can be another way to obtain such nonplanar configurations; this has not been explored in the literature. In this work, we focus on axial growth coupled with rotation of crosssections. This can lead to nonplanar configurations if the material of the rod exhibits some sort of twistextension coupling, simply with a boundary condition that arrests relative axial rotation at the ends. We present a homogeneous growth model for special Cosserat rods with two controls, one for lengthwise growth and the other for rotations. This is explored in greater detail for straight rods with transverse hemitropy and helicalmaterial symmetry by introducing the assumption of symmetry preserving growth to account for the microstructure. The example of a guidedguided rod possessing chiral material symmetry is considered to illustrate the occurrence of outofplane buckling at certain stages of growth (or atrophy). These solutions obtained are flip symmetric and chiral. A complete mirroring of the rod, including both growth and constitutive properties gives a solution with opposite chirality, under the same deformation. Endtoend distance for different combinations of growth and material chiralities are examined to understand the effect of twisting growth on the constitutive twistextension coupling.
Preprint: Pradhan, S.P. and Saxena, P., 2020. Buckling of chiral rods due to coupled axial and rotational growth.arXiv preprint arXiv:2009.02037
CT17 Solid Mechanics
Nils Prigge (ETH)
Embedding calculus and automorphisms of manifolds
Considering the space of diffeomorphisms of a closed manifold as the space of self-embeddings, we can study it using the homotopy theoretic approximations from embedding calculus. I will discuss this approach as well as some recent advances, and I will focus on how we might detect the difference between the approximation and the space of diffeomorphisms using classical invariants of fibre bundles.
BMC03 Topology
Christopher Prior (Durham)
Wavelet decompositions of helicity
Fourier decompositions of magnetic helicity have been used to provide information on the cascade to small scales of magnetic topology in resistive MHD systems, such as Dynamo and driven turbulence models. The helicity and energy of the magnetic field at each Fourier scale are shown to be strongly linked and, in homogeneously driven turbulence, one can even find a scale-based linear decomposition of the energy into its helicity and a correlation tensor of the field. We have recently derived a set of similar tools for relating the helicity and energy content of a magnetic field using wavelet decompositions. This has the relative advantages over Fourier analysis of being applicable in highly inhomogeneous systems as well as providing information about the toology-energy relationship in localised regions of space; highly advantageous in, say, Dynamo models, where helicity must be spatial transported away from the region of dynamo formation. I will introduce these tools and give a short practical guide to methods for applying them.
CT11 Magnetohydrodynamics
Ali Raad (Glasgow)
Existence and Uniqueness of Cartan Subalgebras in Inductive Limit C*-Algebras
Cartan subalgebras of C*-algebras have an essential role in building a bridge between C*-algebras on the one hand, and topological dynamics and geometric group theory, on the other. Hence it is natural to ask when a C*-algebra has a Cartan subalgebra, and if so, to what extent it is unique.
I will begin by introducing these objects, together with examples. After defining inductive limits of C*-algebras, I will discuss recent work which shows that AF-algebras have unique canonical Cartan subalgebras, whilst AI-algebras do not.
Poster
Anthony Radjen (Nottingham)
Asymptotic Solutions of Maxwell's Equations via Generalised Friedlander-Keller Ray Expansions
The standard approach to applying ray theory to solving the Helmholtz and Maxwell equations in the large wave-number limit involves seeking solutions which have (i) an oscillatory exponential with a phase term that is linear in the wave-number and (ii) has an amplitude spectrum expressed in-terms of inverse powers of that wave-number. The Friedlander-Keller modification includes an additional power of this wave-number in the phase of the wave structure, and this additional term is crucial when analysing certain wave phenomena such as creeping and whispering gallery wave propagation. A generalisation of these phenomena, including scattering by perturbed boundaries or a displaced point source, show that an extension of the Friedlander-Keller modification is needed by including even more terms in the phase of the wave structure, and our purpose here is to outline this methodology via examples.
Poster
Jacqui Ramagge (Durham)
What can an algebraist bring to Operator Algebras?
Having asked some colleagues about what they would like to hear about, it appears that it would be useful to give my perspective on self-similarity and algebraic techniques in Operator Algebras. Most of this will not be new although some of it is not yet published. Hopefully all of it will be comprehensible.
BMC04 Operator Algebras
Ariel Ramirez Torres (Glasgow)
A fractional approach for the di usion of species in growing tumours
The diffusion of chemical species, such as nutrients, plays a key role in the growth of a tumour. In general, Fick's law of diffusion is assumed for the description of the species evolution, but recent experimental studies have offer different indications. Based on this evidence, we aim to highlight and study the influence of a non-local type of diffusion acting in an avascular tumour. In particular, we propose to describe these non-local interactions be means of the tools that Fractional Calculus [1] offer. Therefore, we consider a diffusion equation for the evolution of the chemical species (in fact, nutrients) that involves derivatives of fractional order. Based on some of the considerations of a previous work [2], we study the non-local diffusion of the chemical species in a biphasic, growing tumour whose internal structure is subjected to the occurrence of transformations. Because of this latter consideration, we end up with a type of non-locality that changes, not only with respect to the visible deformation of the tumour but also with respect to the internal dynamics [3]. We consider a benchmark problem, and the numerical simulations reveal the relevance of embracing a non-local framework.
References 1. T. M. Atanackovi¢, S. Pilipovi¢, B. Stankovi¢, D. Zorica (2014) Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley-ISTE Ltd. 2. S. Di Stefano, A. Ramírez-Torres, R. Penta, A. Grillo. (2018) Self-influenced growth through evolving material inhomogeneities. International Journal of Non-Linear Mechanics, 106:174-187. 3. A. Ramírez-Torres, S. Di Stefano, A. Grillo (2021) Influence of non-local diffusion in avascular tumour growth. Mathematics and Mechanics of Solids. doi:10.1177/1081286520975086
MS18 Growth and Remodelling in Soft Tissues
David Rand (Warwick)
Geometry, information and genetics
Can we persuade biologists to take seriously the ideas expressed in Rene Thom's famous book Stabilité structurelle et morphogénèse. This was meant to lay out a new approach to morphogenesis and developmental biology and contained much more than what is now understood as Catastrophe Theory. Although Thom was in discussion with C H Waddington, a revered biologist, its deeply philosophical mathematical approach is very far from how biologists think and to take it seriously they need to see how to profit from such an approach. In my opinion Thom also made one serious philosophical mistake: in his Field’s medal autobiography Thom states “… catastrophe theory is dead … it died of its own success … as soon as it became clear that the theory did not permit quantitative prediction”. One of my aims is to show that, on the contrary, combined with other geometrical and dynamical systems theory and advanced statistics, it does permit quantitative prediction in precisely the area where Thom was most interested and this is what is needed to sell it to biologists. Moreover, today we have a huge advantage in increased understanding of genetics and hugely powerful biological data technologies. I will try and convince the audience that new approaches based on geometry, dynamics and information which combines qualitative and quantitative approaches and are very much in tune with Thom's viewpoint can provide new insights into biology, particularly developmental biology and genetics. This is based on joint work with many people, notably Eric Siggia (Rockefeller), James Briscoe (Crick), Merixell Saez (Warwick & Crick), Elena Camacho Aguilar (Rice), Robert Blassberg (Crick), Michael White (Manchester) and Andrew Millar (Edinburgh).
Lighthill Lecture
Dhruv Ranganathan (Cambridge)
4/2 ways of counting curves in a pair
I will discuss the geometry surrounding a simple question: how does one understand the space of rational plane curves with tangency conditions along two distinct lines? There are (probably more than) two very sensible sounding ways of studying this problem: via orbifolds and via logarithmic structure. We’ll see why these give rise to different answers. I’ll then try to explain how an amazing formula of Aluffi, the intersection theory canon of Fulton-Macpherson, and a bit of tropical geometry tell us how to think about this and much more. This is joint work with Navid Nabijou (Cambridge).
BMC07 Algebraic Geometry
Daniel Ratliff (Loughborough)
Entirely Out Of Character? Moving Frames in Dispersive Dynamics
In the evolution of nonlinear waves, localised structures and defects can form and persist, even within stable waves. One way that their formation can be understood is by using the Whitham Modulation equations (WMEs), a dispersionless set of quasilinear PDEs. However, a persistent problem is how to regularise this system via the inclusion of dispersive effects to prevent the emergence of multivalued wave quantities. Surprisingly, it transpires that such features already lurk within the WMEs whenever they are hyperbolic – one merely waits long enough in a suitable moving frame. This takes the form of the Korteweg – de Vries (KdV) equation, and is universal in the sense that its coefficients are tied to abstract properties of the original Lagrangian.
This leads to a more general question – can the properties of the characteristics be used to infer the resulting dynamics? This talk confirms this, and the connection between established concepts in hyperbolic systems (such as the Hamiltonian-Hopf bifurcation and linear degeneracy) and some well-known nonlinear dispersive equations, such as the Two-Way Boussinesq and modified KdV equations, are made.
MS23 Dispersive hydrodynamics and applications
Clare Rees-Zimmerman (Cambridge)
Modelling diffusiophoresis: a promoter of stratification in drying films
Stratification in drying films – how a mixture of differently-sized particles arranges itself upon drying – is examined. It is seen experimentally that smaller particles preferentially accumulate at the top surface, but it is not understood why. Understanding this could allow the design of formulations that self-assemble during drying to give a desired structure. Potential applications are across a wide range of industries, from a self-layering paint for cars, to a biocidal coating in which the biocide stratifies to the top surface, where it is required.
On the basis of diffusional arguments alone, it would be expected that larger particles stratify to the top surface. However, other physical processes, including diffusiophoresis, may also be important. By deriving transport equations, the magnitude of different contributions can be compared, and numerical solutions for the film profile are produced. Asymptotic solutions are derived for the film profile in the high Péclet number (fast evaporation compared to diffusion) regime.
Diffusiophoresis is the migration of particles along a concentration gradient of a different solute species. A particular diffusiophoresis mechanism that has been hypothesised to cause small-on-top stratification is an excluded volume effect. This work probes the significance of this type of diffusiophoresis: to the diffusional model, a diffusiophoresis term is added that can be varied in strength. For hard spheres, it is predicted that diffusiophoresis counteracts the effect of diffusion, resulting in approximately uniform films. When the diffusiophoresis strength is increased, the small particles are predicted to stratify to the top surface. This suggests that diffusiophoresis does contribute to experimental observations of small-on-top stratification, but it might not be the only promoting factor.
CT01 Viscous Fluid Dyamis 1
James Daniel Reilly (Strathclyde)
Dynamics of Coating Flow on Rotating Circular and Elliptical Cylinders
Coating the exterior of an object in a layer of fluid is a fundamental problem in fluid mechanics and occurs in many industrial processes. Perhaps the most well-known example of this is the coating of a rotating horizontal circular cylinder with a thin film of fluid, which was studied in the pioneering papers by H. K. Moffatt (J. de Mécanique 16 1977, 651-673) and V. V. Pukhnachev (J. Appl. Mech. Tech. Phys. 18 1977, 344-351). Since then, this problem has been extended to incorporate a variety of other physical effects (e.g. surface tension and electrical field effects) and remains a rich area of research. While the coating of a circular cylinder has been well studied in recent years, there has been almost no work on non-circular cylinders and, in many applications (such as the coating of chocolate bars and orthopaedic implants), the substrate may not be perfectly circular. Two dimensional flow on the surface of a rotating elliptical cylinder was first studied by R. Hunt (Numer. Methods. Partial Differ. Eqs. 24 2008, 1094-1114) and more recently by W. Li et al. (Phys. Rev. Fluids 2 2017, (9) 094005). In the present work we will use lubrication theory to derive and analyse a reduced model for thin-film coating flow on a rotating elliptical cylinder. This model retains the essential physics inherent in the full problem, but is much less computationally expensive than Direct Numerical Simulation (DNS). Preliminary results show that even a slight eccentricity can cause a radical difference in the behaviour compared to the perfectly circular case. We also examine the novel behaviours which the elliptical shape can give rise to, such as film rupture at the ends of the cylinder.
Poster
Jean Reinaud (St Andrews)
Breaking of baroclinic tori of potential vorticity
Large scale oceanic and atmospheric flows are strongly influenced by the background planetary rotation as well as as the stable density stratification of the fluid. When these two effects are dominant, the flow dynamics can be fully described by the single scalar quantity: the potential vorticity (PV). The latter is materially conserved in absence of diabatatic effects and often negligle frictional effects. Eddies in the oceans can be seen as compact volumes of PV. The contribution of meso-scale eddies to mass transport in the oceans is comparable to the wind-driven or the thermohaline circulations, hence such structure play key roles in the oceans. Hetons are an example of baroclinic eddies which allow to transport quantities over large distances. Hetons are structures consisting of two eddies rotating in opposite directions (a cyclone and an anti-cyclone) lying at different depths. One possible mechanism of formation of hetons is the breaking of larger distributions of PV such as PV rods and PV tori. We discuss the stability and the breaking of baroclinic tori of PV and the subsequent formation of diverging hetons, in the so-called quasi-geostrophic regime.
CT02 Geophysical Fluid Dynamics
Thomas Rey (Lille)
On equilibrium preserving spectral methods for solving the Boltzmann equation
Different approaches are generally used to tackle kinetic equations numerically: deterministic methods, such as finite volume, semi-Lagrangian and spectral schemes, and probabilistic methods, such as Direct Simulation Monte Carlo schemes. In this talk, we shall give an overview of the spectral methods for solving the Boltzmann equation, emphasizing on how these methods can be modified to be structure preserving approximation of the Boltzmann equation. This is a joint work with L. Pareschi.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Davide Riccobelli (Milan)
Role of tissue surface tension in brain organoid morphogenesis
Understanding the mechanics of brain embryogenesis can provide insights on pathologies related to brain development, such as issencephaly, a genetic disease which causes a reduction of the number of cerebral sulci. Recent experiments on brain organoids have confirmed that gyrification, i.e.~the formation of the folded structures of the brain, is triggered by the inhomogeneous growth of the peripheral region. However, the rheology of these cellular aggregates and the mechanics of lissencephaly are still matter of debate.
In this work, we develop a mathematical model of brain organoids based on the theory of morpho-elasticity. We describe them as non-linear elastic bodies, composed of a disk surrounded by a growing layer called cortex. The external boundary is subjected to a tissue surface tension due to intercellular adhesion forces. We show that the resulting surface energy is relevant at the small length scales of brain organoids and affects the mechanics of cellular aggregates. We perform a linear stability analysis of the radially symmetric configuration and we study the post-buckling behaviour through finite element simulations.
We find that the process of gyrification is triggered by the cortex growth and modulated by the competition between two length scales: the radius of the organoid and the capillary length generated by surface tension. We show that a solid model can reproduce the results of the in-vitro experiments. Furthermore, we prove that the lack of brain sulci in lissencephaly is caused by a reduction of the cell stiffness: the softening of the organoid strengthens the role of surface tension, delaying or even inhibiting the onset of a mechanical instability at the free boundary.
MS18 Growth and Remodelling in Soft Tissues
Scott Richardson (Glasgow)
Modelling the coronary circulation
Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid-structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre-reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three-phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid.
We benchmark this approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry as considered in previous work. With this framework, we also simulate the poroelastic dynamics of a three-dimensional left ventricle, in which the myocardium is described by the Holzapfel--Ogden law.
Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously, where we find that the poroelastic LV behaves differently from the hyperelastic LV model.
MS27 Mathematical and Computational Modelling of Blood Flow
Martin Richter (Nottingham)
Convergence Properties of Transfer Operators for Billiards with a Mixed Phase-Space
We analyse the convergence properties of a ray-tracing approach to transfer operators. The investigation focuses on a two-dimensional Hamiltonian system with a mixed-phase space, i.e. coexisting integrable and chaotic dynamics. More precisely, we focus on a two-dimensional billiard domain in which the corresponding wave problem has Dirichlet boundary conditions. As we focus on mid- to high-frequency regimes, we construct the transfer operator by means of a ray-tracing approach. We then solve the propagation problem dynamically and investigate the rate of convergence. We accompany this analysis with an investigation of the dynamics in phase space in terms of the associated boundary map in Birkhoff-coordinates and investigate spectral properties of the transfer operator. We compare our findings with recent proofs carried out for a circular domain and conclude with an outlook about its applicability for real-world problems.
CT13 Dynamical Systems
Alice Rizzardo (Liverpool)
New examples of non-Fourier-Mukai functors
Functors between the derived categories of two smooth projective varieties are a fundamental object of study. Almost all known such functors are so-called Fourier-Mukai: roughly speaking, they are well-behaved with respect to the geometry of the varieties. They admit a lift to a functor between the enhancements of the two derived categories. The first example of a non-Fourier-Mukai functor was given by myself and Van den Bergh in 2015. I will show that this is not a pathological example by providing a way to construct a non-Fourier-Mukai functor from the derived category of any smooth projective variety of dimension greater or equal to 3 admitting a tilting bundle. This is joint work with Theo Raedschelders and Michel Van den Bergh.
BMC07 Algebraic Geometry
Joe Roberts (Oxford)
Modelling the Carding of Recycled Carbon Fibre
The many potential applications and properties of carbon fibre mean that the demand for it has increased in recent years. This means that the amount of waste carbon fibre is increasing. This waste can be recovered and turned into non-woven materials for use in industry. One step in this process is the carding of carbon fibres using carding machines, which are also used in the textile industry. Carding machines consist of a set of toothed rollers of different sizes, moving in different directions and at different velocities, with the aim of producing a web of aligned fibres. In this study, a continuum model is derived for carbon fibres moving through a carding machine, considering different regions of the machine. We examine properties such as the density and order of fibres through the machine, and look at the role of hooks in the combing of the fibres. The aim of this work is to make the process of producing a web of aligned fibres more efficient by examining the properties of the machine.
Poster
Jonna Roden (Edinburgh)
PDE-Constrained Optimization for Multiscale Particle Dynamics
There are many industrial and biological processes, such as beer brewing, nano-separation and bird flocking, which can be described by integro-PDEs. These PDEs define the dynamics of a particle density within a fluid bath, under the influence of diffusion, external forces and particle interactions, and often include complex, nonlocal boundary conditions.
A key challenge is to optimize these types of processes. For example, in nano-separation, it is of interest to determine the optimal inflow rate of particles (the control), which leads to high separation of the particles (the target), at a minimal financial cost. Mathematically, this requires tools from PDE-constrained optimization. A standard technique is to derive a system of optimality conditions and solve it numerically. Due to the nonlinear, nonlocal nature of the governing PDE and boundary conditions, the optimization of multiscale particle dynamics problems requires the development of new theoretical and numerical methods.
I will present the system of nonlinear, nonlocal integro-PDEs that describe the optimality conditions for such an optimization problem. Furthermore, I will introduce a numerical method, which combines pseudospectral methods with a multiple shooting approach. This provides a tool for the fast and accurate solution of these optimality systems. Finally, some examples of future industrial applications will be given. This is joint work with Ben Goddard and John Pearson.
CT05 Optimisation
Colva Roney-Dougal (St Andrews)
Finite simple groups and computational complexity
This talk will describe connections between structural results about the finite simple groups and the complexity of computational algorithms for permutation groups.
The first part of the talk will survey both old and very new results on the minimal number of generators and the base size of a permutation group, two invariants which influence the complexity of the vast majority of permutation group algorithms.
After this, we shall introduce the complexity class NP, and discuss group-theoretic questions for which there is no known polynomial time solution. In particular, we shall present a new approach to computing the normaliser of a primitive group G in an arbitrary subgroup H of S_n. One key tool is Babai’s recent breakthrough on the graph isomorphism problem, and Helfgott's improvement of Babai's bound.
BMC Morning Speaker
Alexander Round (Open University)
Oscillatory behaviour between solitary pulses on falling liquid films
A liquid film flowing down an inclined plane is an example of a convectively unstable open-flow hydrodynamic system with a rich variety of spatiotemporal structures. At the latest stage of the evolution, the film surface is dominated by interacting solitary pulses, which under certain conditions may form bound states, i.e. a group of pulses travelling with the same speed. In this work, we study strong interactions between these pulses, which arise when they are sufficiently close to each other. By making use of time-dependent computations of a low-dimensional model, and bifurcation theory, we observe and quantify the emergence of oscillatory dynamics that strongly depends on the physical parameters of the system. In particular, our results show that as the Reynolds number is increased, the separation length between pulses goes from damped oscillatory states to a limit-cycle through a Hopf bifurcation.
Poster
Malena Sabate Landman (Bath)
Iteratively Reweighted Flexible Krylov methods for Sparse Reconstruction
Krylov subspace methods are powerful iterative solvers for large-scale linear problems, such as those arising in data science and inverse problems in imaging. In this talk I will presents a new algorithm to find a sparse solution for such problems based on $\ell_2$-$\ell_1$ regularization, that is approached by partially solving a sequence of quadratic problems using a flexible Krylov scheme. This algorithm is built upon a solid theoretical justification that guarantees that the sequence of approximate solutions converges to the solution of the original problem, but it has the advantage of building a single (flexible) approximation (Krylov) Subspace that encodes regularization through variable ``preconditioning''. The performance of this algorithm will be shown through a variety of numerical examples. This is a joint work with Silvia Gazzola (University of Bath) and James Nagy (Emory University).
MS02 Mathematics for Data Science
Mario Sandoval (Autonoma Metropolitana)
’Maxwell-Boltzmann’ velocity distribution for noninteracting active matter
We theoretically and computationally find a Maxwell-Boltzmann-like velocity distribution for noninteracting active matter (NAM). To achieve this, mass and moment of inertia are incorporated into the corresponding noninteracting active Fokker-Planck equation (NAFP), thus solving for the first time, the underdamped scenario of NAM following a Fokker-Planck formalism. This time, the distribution results in a bimodal symmetric expression that contains the effect of inertia on transport properties of NAM. The analytical distribution is further compared to experiments dealing with vibrobots. A generalization of the Brinkman hierarchy for NAFP is also provided and used for systematically solving the NAFP in position space. This work is an important step toward characterising active matter using an equivalent non-equilibrium statistical mechanics.
CT 09 Mathematical physics
Graeme Sarson (Newcastle)
MHD in the supernova-driven interstellar medium with cosmic rays
The interstellar medium (ISM) of galaxies is a distinctive MHD environment in many ways. The dominant driving mechanism for small-scale plasma flows -- the expansion of supernovae (SNe) -- effectively acts as a random forcing. The plasma is extremely compressible, and is subject to radiative cooling, separating into populations of cold, warm and hot gas. These processes occur at very high magnetic Prandtl number, so that magnetic fields persist to very small scales. And the medium is also occupied by energetically significant amounts of cosmic rays (CR), which interact with both magnetic field and plasma, significantly contributing to the large-scale flows and structures. We discuss recent simulations bearing on some of these distinctive aspects.
The supernova forcing is irrotational in nature, but the flow develops significant amounts of vorticity and helicity, contributing to large-scale dynamo action. Simulations in a gravitationally stratified and rotating shearing box show that the vorticity production is principally related to the baroclinicity of the flow, especially in regions of hot gas. The net helicities produced by rotation and shear are of opposite signs for galactic rotation laws, with solar-neighbourhood parameters resulting in the near cancellation of total net helicity. The excitation of oscillatory mean flows is also found, interpreted as a signature of the anisotropic-kinetic-alpha (AKA) effect.
MHD simulations usually incorporate CR dynamics via an advection–diffusion equation for CR energy density, with anisotropic (magnetic-field-aligned) diffusion, and with the diffusive flux obeying Fick’s law. We show that a
non-Fickian prescription of CR diffusion can be calibrated to match test particle simulations with great accuracy, and that an appropriate choice of the diffusion tensor can account effectively for the unresolved (subgrid) scales of the magnetic field. We discuss the effects of including CR in this way within MHD simulations of the SN-driven ISM.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
Alan Scaramangas (City, London)
Aposematic signalling in prey-predator systems: determining evolutionary stability when prey populations consist of a single species
Aposematism is the signalling of a defence for the deterrence of predators. Our research focuses on aposematic organisms that exhibit chemical defences, which are usually signalled by some type of brightly coloured skin pigmentation; notably, this is the case for poison frog species of the Dendrobatidae family, although our treatment is likely transferable to other forms of secondary defence. This setup is a natural one to consider and opens up the possibility for robust mathematical modelling: the strength of aposematic traits (signalling and defence) can be unambiguously realised using variables that are continuously quantifiable, independent from one another and which together define a two-dimensional strategy space (right, upper-half plane). We develop a comprehensive mathematical model and explore the joint co-evolution of aposematic traits within the context of evolutionary stability. Even though empirical and model-based studies are conflicting regarding how aposematic traits are related to one another in nature, the majority of works allude to a positive correlation. We suggest that both positively and negatively correlated combinations of traits can achieve evolutionarily stable outcomes and further, that for a given level of signal strength there can be more than one optimal level of defence. Our findings are novel and relevant to a sizeable body of physical evidence, much of which could, until presently, not be addressed in terms of a single, well-understood mechanism.
Poster
Simon Schmidt (Saarbruecken)
On the quantum symmetry of distance-transitive graphs
To capture the symmetry of a graph one studies its automorphism group. We will talk about a generalization of automorphism groups of finite graphs in the framework of Woronowicz's compact matrix quantum groups. An important task is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative. We will see that a graph has quantum symmetry if its automorphism group contains a certain pair of automorphisms. Then, focussing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute and deduce that several families of distance-transitive graphs have no quantum symmetry.
BMC04 Operator Algebras
Bernd Schroers (Heriot-Watt)
The hidden geometry of magnetic skyrmions
This talk is about a model for topological defects in planar ferromagnetic materials, and the geometrical techniques which can be used to obtain infinitely many exact static solutions for certain choices of coupling constants. The defects, called magnetic skyrmions in this context, are widely studied in physics because of their potential role in future magnetic information storage. The applicability of techniques from complex geometry and gauge theory is surprising, and leads to interesting links with the theory of gravitational lensing. Finally, the dynamics of magnetic skyrmions in response to an applied current, which is of practical interest, also contains unexpected geometry and can be understood in terms of quaternionic Moebius transformations.
BMC Morning Speaker
Dirk Schuetz (Durham)
A Scanning Algorithm for Odd Khovanov Homology
We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. The main difficulty comes from the sign assignments in the cochain complex, which are not local in the odd theory. To deal with this we use a mapping cone construction instead of a tensor product to handle the gluings of tangles.
The algorithm has been implemented in a computer program, and we can also use it to make efficient calculations for a concordance invariant that was recently introduced by Sarkar-Scaduto-Stoffregen.
BMC03 Topology
Linus Schumacher (MRC Edinburgh)
Dissecting the dynamics of heterogeneous stem cell populations through Bayesian inference and model comparison
The notion of cell states is increasingly used to classify cellular behaviour in development, regeneration, and cancer. This is driven in part by a deluge of data comprising snapshots of cell populations at single-cell resolution. Yet quantitative predictive models of cell states and their transitions remain lacking. Such models would allow us to fully leverage datasets to gain a quantitative understanding of cell state transitions; and help to optimise the production of a cell types from pluripotent stem cells in vitro.
First, we explore systematically to what extent cell state transition rates can be inferred quantitatively from snapshot data labelling subpopulations. We investigate early cell fate decisions in pluripotent embryo-like stem cells (Tsakiridis et al., 2014). We build a minimal mathematical model for the transitions between these states in a growing cell colony. We adopt a Bayesian inference approach to infer cell state transition rates and their uncertainties. With this data-driven modelling approach we identify statistical dependencies between genes indicating regulatory interactions through Bayesian model comparison. This method is be generally applicable to binary gene expression data from cell populations, and can be extended to analyse single-cell level clonal data. When analysing data from growing clones not at steady state, we further use Approximate Bayesian Computation model comparison to infer whether cell divisions and state transitions are coupled.
MS06 Tracking cellular processes through the scales
Nikolaos Sfakianakis (St. Andrews)
Cell migration and local cancer invasion in 2D- and 3D-environments
The ability to locally degrade the Extracellular Matrix (ECM) and interact with the tumour microenvironment is a key process distinguishing cancer from normal cells, and is a critical step in the tumour metastasis. The tissue invasion involves the coordinated action of the cancer cells, the ECM, the Matrix Degrading Enzymes, and the Epithelial-to-Mesenchymal transition (EMT).
In this talk, we present a 2- and 3D mathematical model which describes the transition from an epithelial invasion strategy of the Epithelial-like Cancer cells (ECs) to an individual invasion strategy for the Mesenchymal-like Cancer cells (MCs); this is a genuinely multiscale and hybrid model of PDEs and SDEs.
MS06 Tracking cellular processes through the scales
Osian Shelley (Warwick)
Transaction tax in a general equilibrium model
In this talk, we consider the effects of a quadratic tax rate levied against two agents with heterogeneous risk aversions in a continuous-time, risk-sharing equilibrium model. The goal of each agent is to choose a trading strategy which maximises the expected changes in her wealth, for which an optimal strategy exists in closed form, as the solution to an FBSDE. This tractable set-up allows us to analyse the utility loss incurred from taxation. In particular, we show why in some cases an agent can benefit from the taxation before redistribution. Moreover, when agents have heterogeneous beliefs about the traded asset, we discuss if taxation and redistribution can dampen speculative trading and benefit the agents, respectively.
CT 15 Statistical and Numerical Methods
Josh Shelton (Bath)
On the structure of parasitic gravity-capillary waves in the small surface tension limit
It is well known observationally that under the action of surface tension, parasitic ripples of small wavelength form on the forward face of a steep propagating gravity wave. We study one aspect of this problem: the structure of periodic travelling gravity-capillary waves in deep water in the small surface tension limit. Fixing the wave energy as an amplitude parameter, a detailed numerical solution space is uncovered in the Froude (wavespeed) vs Bond (surface-tension) plane for small values of the Bond number. This presents numerical evidence for countably infinite sets of solutions accumulating to an interval of values of the Froude number. Some of these solutions exhibit parasitic structure. The bifurcation connecting two adjacent sets of these solutions shifts their point of symmetry, which has previously been misinterpreted as a discontinuity in the solution branch. The magnitude of these ripples becomes exponentially small in the Bond number, necessitating the use of exponential asymptotics to describe their form analytically. We will also introduce this theory, which relates the functional form of the ripples to the singularities of the analytic continuation of the Stokes gravity wave.
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Rebecca Shipley (UCL)
Multiscale Models of Tumour Fluid and Drug Distribution: Integrating Imaging and Computation
Understanding how drugs and other therapeutics are delivered to a diseased tissue, and their subsequent spatial and temporal distribution, is a key factor in the development of effective, targeted therapies. Capturing the physiological variation in complete, intact tissue specimens is particularly useful in tumours, which can be highly heterogeneous, both between tumour types and even within individual tumours. However, it is virtually impossible to quantify drug delivery across whole tumour samples through experimental imaging alone.
Here we propose the development of multiscale models of tumour fluid and drug distribution, integrated with high resolution microstructure imaging, to gain insight into this problem. We present a suite of imaging modalities and data sets which provide both high-resolution microstructure data for whole tumours, as well as tissue-scale perfusion information. We present a series of computational models that capture transport phenomena across these length scales, and are calibrated and validated against the modelling data. Finally, we present modelling predictions which provide insight into treatment strategies for a range of cancer therapeutics.
Plenary (EMS)
Jemma Shipton (Exeter)
Modelling freely decaying shallow water turbulence with compatible finite element methods.
We present results from recent compatible finite element simulations of freely decaying shallow water turbulence. Compatible finite element methods are a form of mixed finite element methods (meaning that different finite element spaces are used for different fields) that allow the exact representation of the standard vector calculus identities div-curl=0 and curl-grad=0. This necessitates the use of div-conforming finite element spaces for velocity, such as Raviart-Thomas or Brezzi-Douglas-Marini, and discontinuous finite element spaces for pressure. The development of these methods for numerical weather prediction has been motivated by the parallel scalability bottleneck of the standard latitude-longitude grid. Cotter and Shipton [2012] demonstrated that compatible finite element discretisations for the linear shallow water equations satisfy the basic conservation, balance and wave propagation properties listed in Staniforth and Thuburn [2012], without the restriction that the grid is orthogonal. The linear equations dictate our choice of finite element spaces; we then need to construct stable and accurate advection schemes to solve the nonlinear equations. Here we focus on the challenging problem of modelling freely decaying turbulence. In this situation, vortices form, interact, and like-signed vortices merge. In the spherical case the flow evolves into zonal bands. We investigate the effect of different spatial and temporal discretisations on the properties of the flow.
CT02 Geophysical Fluid Dynamics
Matthew Shirley (Oxford)
Heat transfer by gas flow in silicon furnaces
In silicon furnaces, channels are observed to form through the solid raw materials and hot gases created by chemical reactions flow through these channels. The gases play a vital role in heating the incoming raw materials, both through diffusive heat transfer and through exothermic chemical reactions between the gas and solid. Understanding the gas flow is therefore crucial to building a realistic model of heat transfer in the furnace, and in turn optimising the furnace's efficiency.
In this talk we present a mathematical model for the laminar flow of compressible gas in such a channel. Using the slenderness of the channel and the small pressure drop driving the flow, we asymptotically obtain a simplified system of equations which we solve to predict how the temperature of both the gas and the surrounding solid varies with depth. We discuss how changing the geometry of the channel, and the operating parameters of the furnace affects the gas flow and thermal distribution.
CT01 Viscous Fluid Dyamis 1
David Sibley (Loughborough)
How ice grows from premelting films and water droplets
In situations where solid ice is in contact with a vapour water phase at temperatures close to the triple point the ice surface is covered by a thin liquid layer of water, whose presence impacts on the dynamics of ice growth. Here, we develop a coupled solid-liquid-vapour system for the case of water where both freezing/melting and evaporation/condensation processes are included. From our mesoscopic model we are able to elucidate a variety of ice growth behaviours dependent on the ambient temperature and vapour pressure, with excellent agreement between an analytic phase-diagram and numerical approaches for the evolution. Our model is able to capture experimentally observed processes such as lateral terrace motion at low saturation and a transition to capture liquid droplets at higher saturations, and also we uncover growth processes beneath the liquid layer such as crater formation in the ice surface.
Poster
Sue Sierra (Edinburgh)
Poisson geometry of the Virasoro algebra
Let W be the Witt Lie algebra of algebraic vector fields on the punctured complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central extension of W, with central generator z. We describe the geometry associated to prime Poisson ideals in Sym(Vir).
We focus first on understanding Poisson primitive ideals: Poisson cores of maximal ideals in Sym(Vir). We classify maximal ideals which have nontrivial Poisson cores and calculate their Poisson cores; using this we show that if \lambda \neq 0 then Sym(Vir)/(z-\lambda) is Poisson simple.
There is a notion of coadjoint orbit of a point in Vir^* = MSpec(Sym(Vir)). Although Vir^* is infinite-dimensional, these coadjoint orbits are all finite-dimensional. We describe them explicitly and also discuss general prime Poisson ideals and implications for the ideal structure of the universal enveloping algebra of Vir and for the representation theory of Vir.
This is joint work with Alexey Petukhov.
BMC01 Algebra and Representation Theory
Samir Siksek (Warwick)
Efficient resolution of Thue–Mahler equations
A Thue–Mahler equation has the form F(X,Y)=p_1^{m_1} \cdots p_r^{m_r} where F is an irreducible homogeneous binary form of degree at least 3 with integer coefficients, and p_1,..,p_r are primes. A standard algorithm due to Tzanakis and de Weger solves Thue–Mahler equations when the degree and the number of primes is small. We give lattice-based sieving techniques that are capable of handling large Thue–Mahler equations. This is joint work with Adela Gherga, Rafael von Känel and Benjamin Matschke.
BMC02 Number Theory
Michael Singer (UCL)
Brilliant Corners
When we first meet functions of two variables, examples such as xy/(x^2+y^2) are introduced, usually to show that a function can be quite bad even though its partial derivatives exist everywhere. This example is bad because if (x,y) goes to zero, then the limit depends on the direction of approach to the origin. On the other hand the function is completely smooth in polar coordinates around the origin.
While this example may appear to be pathological, constructed specifically to scare students, such direction-dependent limits are very frequently encountered in the study of PDE problems in non-compact settings or when there is small parameter giving a singular limiting problem.
The title of the talk refers to a circle of ideas, pioneered by Richard Melrose, in which manifolds with corners are used systematically to study such problems. I aim to survey these ideas through geometric examples which illustrate how brilliant corners can be.
BMC Morning Speaker
Steven Sivek (Imperial)
Framed instanton homology and Dehn surgery
Framed instanton homology is a 3-manifold invariant which is usually very difficult to compute. However, it turns out that for any knot K in S^3, the homologies of all Dehn surgeries on K are determined by a single pair of integers, one of which is even a smooth concordance invariant. In this talk we’ll discuss how a new symmetry property of cobordism maps in framed instanton homology constrains the geography of these invariants, with applications to homology cobordism and to the Dehn surgery realization problem for many rational homology 3-spheres.
BMC03 Topology
Piotr Słowiński (Exeter)
Distance-based analysis of covariance matrices; Riemannian geometry and Frobenius norm.
Covariance matrices are a popular tool for analysing multivariate data. Their applications vary from neuroscience to economics. I present a pipeline for analysing covariance matrices using distance-based statistics: bias corrected distance correlation (Székely & Rizzo, 2013) and interpoint distance test (Marozzi, Mukherjee, & Kalina, 2020). Using human movement data, I demonstrate and explain how the results of the analysis depend on the choice of metric. Frobenius distance differentiates the data depending on the speed of movement while exploiting Riemannian geometry of the positive semi-definite matrices allows to tell apart two age groups of the participants. When studying covariance matrices, different metrics provide complementary insights into analysed data.
Poster
Simon Smith (Lincoln)
Local to global behaviour of groups acting on trees
Groups acting on trees play a fundamental role in the theory of groups.Bass--Serre Theory, and in particular the notion of a graph of groups, is a powerful tool for decomposing groups acting on trees, but its usefulness for constructing non-discrete groups acting on trees is, in some situations, severely limited. Such groups play an important role in the theory of infinite permutation groups, and the theory of t.d.l.c. groups, as they are a rich source of examples of simple groups. For these groups proving simplicity is often tricky,and relies on the group satisfying some sort of independence property, usually Tits' independence property (P).
An alternative, but complementary, approach to the study of groups acting on trees has recently emerged based on local actions, that is, the action of a vertex stabiliser on the neighbouring vertices. In many situations this alternative is better suited to constructing non-discrete groups acting on trees. The beginnings of this approach can be found in a 2000 paper of M.~Burger and Sh.~Mozes, in which the authors use this `local-to-global' approach to construct an interesting class of (virtually) simple t.d.l.c. groups acting on trees, all with Tits independence property (P). The majority of new constructions of compactly generated simple t.d.l.c. groups have used ideas inspired by this work.
More recently, the Burger--Mozes construction has been generalised (by the speaker) to obtain a new product (called the box product) of permutation groups, and this was used to prove that there are $2^{\aleph_0}$ isomorphism types of non-discrete, compactly generated, simple t.d.l.c. groups. It also formed an integral part of the recent classification of subdegree-finite primitive permutation groups.
This generalisation also uses a `local-to-global' approach to construct groups acting on trees with Tits independence property (P).
In joint work with Colin Reid, we have developed a general method for describing and classifying all actions of groups on trees with property (P). This is done using an object called a local action diagram, akin to a graph of groups, but for local actions. Our work can be seen as a `local action' complement to Bass-Serre theory. Under this framework, the Burger--Mozes construction corresponds to a local action diagram consisting of a single vertex with a set of loops, each with its own reverse, and the box product construction corresponds to a local action diagram consisting of a pair of vertices and no loops. Any connected graph can form the basis of a local action diagram and each local action diagram gives rise to a unique (up to conjugacy) group with property (P).
Furthermore, for a group $G$ with property (P), one can easily determinewhether or not $G$ has certain properties (e.g. compact generation and simplicity) directly from its local action diagram.
BMC09 Groups
Efthymios Sofos (Glasgow)
Prime values of integer polynomials and random Diophantine equations
Schinzel's hypothesis is a central conjecture in number theory which states that integer polynomials satisfying the obvious necessary assumptions represent primes. It is completely open in all cases except when the polynomial has degree 1 . In joint work with Alexei Skorobogatov we settle the conjecture in 100% of the cases (when polynomials are ordered by height of coefficients). Furthermore, we explore consequences for integer solutions to random Diophantine equations.
BMC02 Number Theory
Mikhail Sokolovskiy (Russian Academy of Sciences)
Interaction of an intrathermocline vortices with a surface synoptic cyclone in a three-layer model of the ocean
Intrathermocline lenses are vortex patches (anticyclonic or cyclonic) localized at intermediate horizons (600–1600 m) of the ocean. They are observed everywhere in the World Ocean, but mainly in the North Atlantic. A typical situation is when anticyclonic and cyclonic lenses are located one next to the other, and both are affected by a large-scale vortex [1]. In this work, we study some mechanisms of interaction of a intrathermoclinic lenses with a synoptic surface vortex. We use a three-layer quasi-geostrophic model with a density stratification in the form of a two-step piecewise constant function that approximates the mean multi-year vertical density distribution of the North Atlantic. Anticyclonic and cyclonic intrathermocline lenses are presented in the form of vortex patches in the middle layer with potential vorticities of negative and positive signs, respectively, and with horizontal scales of the order of the Rossby deformation radius. The surface cyclone, having a radius of 4-6 times larger, is presented by a vortex patch in the upper layer. The bottom layer is considered as passive.
MS16 Eddies in Geophysical Fluid Dynamics
Josephine Solowiej-Wedderburn (Surrey)
A mathematical model for a contractile mechanosensory mechanism within cells
Cells interact with their environments through a variety of chemical and physical signalling mechanisms. It is becoming increasingly clear that physical force and the mechanical properties of their microenvironment play a crucial role in determining cellular behaviour and coordination. Understanding these differences is crucial for tissue engineering applications and to determine how the mechanical microenvironment may affect, for example, cancer growth and invasion. We use a continuum elasticity-based model with an active contractile component to describe the mechanosensory mechanism of a cell or cell layer adhered to a substrate. The model context focuses on the most common biophysical experimental set-ups investigating cellular contractility, using experimentally determined knowledge of the mechanical response of the designed substrates to infer the cell-generated force from observed substrate deformations. The mathematical model is analysed and solved using both analytical approaches (exploiting approximations and symmetry arguments) and Finite Element Methods. We use the model to explain observed cellular adaptations to changes in the mechanical properties of the underlying gel. In particular, we consider the distribution of adhesion and contractility throughout a cell. For experimentally realistic distributions of adhesion points, the model is capable of recreating cell shapes and deformations that are consistent with those experimentally observed. Furthermore, energy considerations are shown to have significant implications for the optimisation of cell adhesion and the organisation of cellular contractility.
CT 16 Mathematical Biology-3
Emma Southall (Warwick)
Evaluation of lead time prediction methods for detecting critical transitions using timeseries data
Early-warning signals are widely used in many fields to anticipate a critical threshold prior to reaching it. A systems undergoes the phenomenon known as critical slowing down as it crosses through a bifurcation. Theory predicts that fluctuations away from the mean will recover more slowly as the system approaches a critical transition. This is key in infectious disease modelling to assess when the basic reproduction number is reduced below the threshold of one.
Recent theoretical advances have shown indicators of critical transitions in epidemiology such as measuring the lag-1 autocorrelation in synthetic disease data. An effective early-warning signal would be able to predict an impending critical transition of this type with a suitable lead time in order to act on the current path of the disease.
We validate several empirical studies which offer lead time predictions for ecological and infectious diseases systems when using this theory practice. Our work highlights several challenges when applying lead time methodologies to simulated models. We find poor specificity, falsely reporting defections of a critical transitions in simulations at steady state.
In this talk we present an extension to these methods and our results show promising potential for calculating early-warning signals of elimination on real-world noisy data.
CT04 Mathematical Biology-1
Douglas C. Speirs (Strathclyde)
Modelling the spatial population dynamics of a marine zooplankton species under climate change
The copepod Calanus finmarchicus is widely distributed over the North Atlantic, and dominates the zooplankton biomass in much of that region. Its geographic distribution has been changing recently, with ocean warming extending its northern limit further into the Arctic. Calanus has a complex life cycle involving numerous life-history stages, including five copepodite stages preceding a final moult into adulthood. The fifth copepodite stage (CV) is capable of sinking to great depth (in excess of 1000m) and entering a torpid overwintering state before rising again to the surface in the spring to reproduce. Since CVs store energy reserves in the form of lipids, this huge vertical migration involves a large transfer of surface production to the deep ocean. Recent studies have quantified this effect, known as the lipid pump, and have shown that the amount of carbon sequestered by C. finmarchicus alone is approximately the same as the annual sinking flux of detrital carbon. So, warming Arctic waters not only have the potential to change its distribution, but also may have an impact on deep ocean carbon sequestration. Modelling these processes in order to improve our predictive capability involves significant challenges in combining both spatial structure and stage structure. Here we present a new model uses a computationally-efficient approach that combines transport derived from Lagrangian particle tracking with an Eulerian population representation, and incorporates recent advances in our understanding of Calanus biology. The model is driven by physical transport, temperature, and food, derived from global coupled physical-biological models. We show that the predicted changes in geographic range of C. finmarchicus are not consistent with previously published studies. In particular, although ocean warming produces an initial range expansion, ultimately population declines occur because of the increased costs of overwintering when deep-water warming occurs. Work done jointly with Robert J. Wilson and Michael R. Heath.
MS13 Mathematical challenges in spatial ecology
Helena Stage (Manchester)
Multi-Scale Superinfection Models in Evolutionary Epidemiology
The study of evolutionary epidemiology is vital to understand and control the spread of anti-microbial resistance, but is inherently challenging because pathogen evolution is driven by forces acting at multiple scales: for example, HIV needs to escape the immune system within a host, but also needs to maintain the ability to be transmitted efficiently between hosts. Time-since-infection models are much more flexible than ODEs if we want to allow for realistic enough aspects of both within- and between-host scales, but capturing the feedback loops between such scales is a formidable challenge.
We will discuss the main technical challenges in developing a general theory for time-since-infection models that allow for superinfection (e.g. multi-strain systems with partial cross-immunity), starting from the problem of characterising the system’s steady states. We will distinguish between the cases when superinfection of the host facilitates the coexistence of two (or more) infections that interact synergistically by fuelling each other’s spread (syndemic), and when these infections hinder each other. We show how in the former case multiple stable steady states are possible, while in the latter case the stable steady state is unique but possibly harder to compute. We discuss the consequent implications for public health control measures.
CT04 Mathematical Biology-1
Ioan Stanciu (Oxford)
Primitive ideals in the affinoid enveloping algebra of a semisimple Lie algebra
I will start by defining the affinoid enveloping algebra of a semisimple Lie algebra and explain the connection with the Iwasawa algebra and the classical enveloping algebra. Next, I will review the characterisation of primitive ideals in the classical enveloping algebra of a semisimple Lie algebra and explain how one can use geometric representation theory to obtain an affinoid version of Duflo's theorem. Finally, I will talk about how a large class of two-sided ideals in the affinoid enveloping algebra is controlled by two-sided ideals in the classical enveloping algebra
BMC01 Algebra and Representation Theory
Brigitte Stenhouse (Open University)
Preparing a mathematical translation: Mary Somerville’s 1831 Mechanism of the Heavens
Mary Somerville’s 1831 work, Mechanism of the Heavens, was widely recognised for its importance in bringing analytical mathematics, and its applications to physical astronomy, to wider attention in early-19th-century Britain. The single volume work was ostensibly a translation of Laplace’s Traité de Mécanique Céleste, which had been published in 5 volumes between 1799 and 1825. One of the many arguments given for the perception of a decline in British mathematics at the time was the small number of British mathematicians sufficiently literate in analysis and algebra to read and understand Laplace’s work. Therefore, when producing her translation Somerville was required to act as both interpreter of the French language, and of the mathematical language and methods employed by Laplace; moreover, she incorporated numerous improvements that had been made since their original publication.
My poster will showcase the changes made by Somerville through a consideration of the work itself alongside contemporary reviews and her correspondence with John Herschel, and investigate translation as a form of mathematical work.
Poster
Greg Stevenson (Glasgow)
Points of cochains on BG and their tangent vectors
Given a prime ideal of a commutative noetherian ring we can attach to it a residue field k and a k-vector space of tangent vectors to the corresponding point in the associated affine scheme. I'll explain how to package this in a homotopy invariant fashion and thus extend these concepts to the cochains on the classifying space of a finite group (amongst other settings). This is based on joint work with Paul Balmer and Henning Krause, and with James Cameron.
BMC03 Topology
Peter Stewart (Glasgow)
Nonlinear Rayleigh--Taylor instability in aqueous foam fracture
A monolayer of gas-liquid foam can exhibit a brittle fracture mode when subjected to a large driving pressure, analogous to brittle fracture of crystalline atomic solids. The brittle crack advances through successive rupture of liquid films, each driven by the Rayleigh--Taylor instability. We analyse the linear stability of a uniform foam lamella accelerated in the direction perpendicular to its interfaces, constructing a reduced approximation for the growth rate of Rayleigh--Taylor instability in the limit of large surface tension. We show that in this limit the asymptotic description can be extended to form a system of one-dimensional partial differential equations which govern the nonlinear growth of the instability and subsequent film rupture. The nonlinear model predicts that, following the onset of instability, the liquid film thins exponentially at its edges and bulges at its centre. As a result the time taken for the film to rupture can be shown to be almost twice that predicted by linear theory alone.
CT08 Industrial fluids
Catharina Stroppel (Bonn)
Tensor products and branching - changes of perspective
Understanding restrictions of group actions to subgroups or tensor products of representations is a classical problem with a long history. Depending on the viewpoint and context one might call it well-understood or vastly unclear. In this talk I like to illustrate how the perspective of attacking such problems changed over the years and indicate how it lead to the development of important new concepts and surprising connections. A new way of drawing classical partitions and standard tableaux for instance allows already to make direct connections with 2-dimensional TQFTs, certain Fukaya categories, supergroups etc. which might have been impossible without a change of perspective.
Plenary
Catharina Stroppel (Bonn)
Verlinde rings and DAHA actions
In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.
BMC01 Algebra and Representation Theory
Raphael Stuhlmeier (Plymouth)
Deterministic wave forecasting with the Zakharov equation
Deterministic wave forecasting aims to provide a wave-by-wave prediction of the free surface elevation based on measured data. Such information about upcoming waves can inform marine decision support systems, control strategies for wave energy converters, and other applications. Unlike well-developed stochastic wave forecasts, the temporal and spatial scales involved are modest, on the order of minutes or kilometres. Due to the dispersive nature of surface water waves, such forecasts have a limited space/time horizon, which is further impacted by the effects of nonlinearity. I will discuss the application of the reduced Zakharov equation, and simple frequency corrections derived therefrom, to preparing wave forecasts. Unlike procedures based on solving evolution equations (e.g. high order spectral method), such corrections entail essentially no additional computational effort, yet show marked improvements over linear theory.
MS26 Recent Advances in Nonlinear Internal and Surface Waves
Bernd Sturmfels (UC Berkeley)
Linear PDE with Constant Coefficients
We discuss algebraic methods for solving systems of homogeneous linear partial differential equations with constant coefficients. The setting is the Fundamental Principle established by Ehrenpreis and Palamodov in the 1960’s. Our approach rests on recent advances in commutative algebra, and it offers new vistas on schemes and coherent sheaves in computational algebraic geometry.
Plenary
Priya Subramanian (Oxford)
Strongly nonlinear theory for the density distribution in soft matter
Density functional theory (DFT) is a microscopic theory that allows us to describe the density distribution in a soft matter system using interparticle interactions and the underlying thermodynamic conditions as input. Prevalent ansaetze to describe density distribution is to either write it as the sum of Gaussians centered at lattice sites or as a Fourier sum of the reciprocal lattice vectors. Close to onset of crystallisation, there can be deviations from the sum of Gaussian description, while on the other hand, when the density distribution has sharp peaks, the Fourier sum can require many terms to be accurate. We show that using the anstaz that the logarithm of the density distribution is a Fourier sum is accurate and more useful than considering either of the above descriptions. This strongly nonlinear framework allows us to compute 3D DFT crystal and quasicrystal solution branches with little effort, both close to onset and when the density distribution has sharp peaks.
MS03 Mathematical aspects of non-equilibrium statistical mechanics
Andrew Sutherland (MIT)
Stronger arithmetic equivalence
Number fields with the same Dedekind zeta function are said to be arithmetically equivalent. Such number fields necessarily have the same degree, discriminant, signature, Galois closure, and isomorphic unit groups, but may have different regulators, class groups, rings of adeles, and idele class groups. Motivated by a recent result of Prasad, I will discuss three stronger notions of arithmetic equivalence that force isomorphisms of some or all of these invariants without forcing an isomorphism of number fields, along with explicit examples and some open questions. These results also have applications to the construction of curves with isomorphic Jacobians (due to Prasad), isospectral Riemannian manifolds (due to Sunada), and isospectral graphs (due to Halbeisen and Hungerbuhler).
BMC02 Number Theory
Louise Sutton (Manchester)
Decomposable Specht modules
The representation theory of the symmetric groups and associated Hecke algebras can be studied via the KLR algebras, whose representations are constructed as graded Specht modules. In this talk, we review a catalogue of results on decomposable Specht modules in level 1 - the study of which began in the 80s by Murphy on Specht modules indexed by hooks for the symmetric group. We move on to discuss recent work in level 2 of the KLR algebras on decomposable Specht modules - namely those indexed by a pair of hooks - in which we describe the structure of large families of semisimple and close-to-semisimple Specht modules.
BMC01 Algebra and Representation Theory
Ben Swallow (Glasgow)
Efficient Bayesian parameter inference for high-dimensional stochastic dynamical biological systems using an approximation
Simulation of non-linear stochastic dynamical models, frequently studied in systems biology, has historically been too slow to enable parameter inference for all but the simplest problems. In order to calculate the probability distributions of these models, approximations to the likelihoods are frequently necessary, but these come with associated drawbacks in long-term accuracy. Recent advances in both stochastic simulation algorithms and efficient Bayesian parameter estimation methodology enable much larger systems, such as those found in cell signalling applications, to be analysed and at least some of their parameters inferred. I will discuss how these recent advances can be used and present results on simulated data from a high-dimensional model of NF-kappaB regulation and a drosophila circadian clock. We study a variety of Markov chain Monte Carlo algorithms under varying system conditions and compare the ability of the algorithms to estimate parameters in these often highly non-identifiable systems.
MS04 Stochastic models in biology informed by data
Angela Tabiri (African Institute for Mathematical Sciences, Ghana)
Femafricmaths: Female African Mathematicians
We are a network of female African mathematicians transforming Africa with mathematics. At Femafricmaths, we interview Female African Mathematicians to highlight the different career options available when you study mathematics. Our videos are available on our YouTube, Facebook and Instagram pages. We also promote the study of mathematics in primary and secondary schools. This comes under the Y3p3maths project which involves volunteers teaching mathematics in person or online. The goal is to support students to study, understand and apply mathematics. Videos of lessons covered during Y3p3maths are available on the Femafricmaths Facebook and YouTube pages. What course do you need to study in order to become an astronaut? Our mentoring for senior high school girls provides guidance to female students on subject to choose when they go to the university. Mathsqueens will be available online to share their career journey with the young girls to guide them in their choice of courses and careers. Our mission is to inspire young people about the diverse career choices available after completing a degree in mathematics through interviews with #mathsqueens and school outreach activities. Our vision is to see young girls become confident to choose a career in mathematics related fields.
Twitter: [@FemAfricMaths](https://twitter.com/femafricmaths)
Outreach Video
Matteo Taffetani (Bristol)
Stokes - Biot coupling in a perfusion bioreactor with inhomogeneous porosity
Authors: Matteo Taffetani (Oxford), Ricardo Ruiz Baier (Monash, Australia) , Sarah Waters (Oxford)
A perfusion bioreactor for tissue growth can be simplified as a fluid-poroelastic system constituted by of two domains, one fluid and one porous, that interface with each others through a single boundary and with a prescribed flux imposed at the inlet. Cells are mechanosensitive entities so the local cellular mechanical environment may be tuned by controlling the geometry of the two phase poroelastic scaffold in terms of spatial distribution of the porosity.
Our modelling approach is motivated by this latter aspect and we describe the porous domain using the equations of linear poroelasticity with the permeability and the shear modulus of the poroelastic material dependent upon the initial porosity field and we determine the impact of the heterogeneity on the kinematic variables (fluid velocity and solid displacement) and on the associated load distribution (pressure and stresses). We first show how the pressure and the strain fields are related to the permeability and the elastic properties solving the proper one dimensional problem and evaluating the the lubrication limit in the small Reynolds number regime. Then we compute the full 2D fields by means of finite element simulations.
CT06 Porous media
Katherine Tant (Strathclyde)
Effective Orientation Mapping of Locally Anisotropic Media from Boundary Measurement Data
In this talk I will examine the inverse problem of reconstructing the spatially varying anisotropic elastic properties of a solid object from transmitted wave measurements made on its boundary. Specifically, I will study the case where the material has a constant density but is constructed from grains which are locally anisotropic and can thus be characterised by their orientation. Given time of flight data measured on the boundary of the object, the reversible-jump Markov chain Monte Carlo method is used to approximate the posterior distribution on grain orientation at each point in the spatial domain, the moments of which can be used to reconstruct an effective material map and comment on the associated uncertainty. The success of the reconstruction is measured by its ability to correct imaging delay laws to produce more focussed characterisations of defects embedded within the object. Although this work was developed with the ultrasonic non-destructive inspection of metals in mind, the methodology has potential applications in seismology and medicine, for example in mapping the multi-phase nature of the Earth’s subsurface or imaging anisotropic fibrous tissue in the breast.
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Aretha Teckentrup (Edinburgh)
Convergence of Gaussian process emulators with estimated hyper-parameters
We start with a short overview of the research area Mathematics for Data Science and the talks in this session.
Then follows a research talk, in which we consider the use of Gaussian process regression to approximate a function of interest. A particular focus is on the case where hyper-parameters appearing in the mean and covariance structure of the Gaussian process emulator are a-priori unknown, and are learnt from the data, along with the posterior mean and covariance. We work in the framework of empirical Bayes, where a point estimate of the hyper-parameters is computed, using the data, and then used within the standard Gaussian process prior to posterior update. Using results from scattered data approximation, we provide a convergence analysis of the method applied to a fixed, unknown function of interest.
MS02 Mathematics for Data Science
Jack Thomas (Warwick)
Tight Binding Models for Insulators: locality of interatomic forces & geometry optimisation
The tight binding model is a minimalistic electronic structure model for predicting properties of materials and molecules. We show that the potential energy surface of this model can be decomposed into exponentially localised site energy contributions, thus providing qualitatively sharp estimates on the interatomic interaction range which justifies a range of multi-scale models. For insulators, we demonstrate that “pollution” of the spectral gap by a point spectrum caused, for example, by local defects in the crystal, only weakly affects the locality estimates. Numerical tests confirm our analytical results. Time permitting, we shall extend our results to a self-consistent (non-linear) tight binding model.
We then describe atomistic geometry relaxation for point defects in the tight binding model. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. Finally, we discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.
This work extends [Chen, Ortner. Multiscale Model. Simul., 2016] and [Chen, Lu, Ortner. Arch. Rational Mech. Anal., 2018] to the case of zero Fermi-temperature as well as strengthening the locality results proved therein.
Joint work with Christoph Ortner and Huajie Chen.
CT17 Solid Mechanics
Alice Thompson (Manchester)
Bubble propagation in modified Hele-Shaw channels: modelling and control
The propagation of a deformable air finger or bubble into a fluid-filled channel with an imposed pressure gradient was first studied by Saffman and Taylor. Assuming large aspect ratio channels, the flow can be depth-averaged and the free-boundary problem for steady propagation solved by conformal mapping. Famously, at zero surface tension, fingers of any width may exist, but the inclusion of vanishingly small surface tension selects symmetric fingers of discrete finger widths. At finite surface tension, Vanden-Broeck later showed that other families of 'exotic' states exist, but these states are all linearly unstable. By including a centred constriction in the channel, multiple modes of propagation can be observed in experiments, including symmetric, asymmetric and oscillatory states, with evidence of a correspondingly rich bifurcation structure.
Depth-averaged modelling is an invaluable tool for understanding this system, allowing us to identify the physical mechanisms underlying each observed propagation mode. With a reduced model, we can calculate steady states, time-dependent evolution, bifurcation tracking and stability analysis. In particular, I will outline our efforts to understand how the system dynamics may be affected by nearby unstable solution branches and how we can accurately predict phenomena such as bubble break up and subsequent multi-bubble interactions from weakly nonlinear stability analysis and fully nonlinear simulations.
Finally, I will discuss a new focus on using feedback control methods to explore and directly observe the unstable states in this system. It is likely that the reduced models do not capture all the details of the unstable states, and indeed may differ even in the bifurcation structure itself. We can avoid the limitations of the model by using control based continuation methods. Here feedback control is used to stabilise states, but the observations are made from the experiment itself. I will discuss some of the particular challenges involved in implementing this method for our system, and how modelling and simulation may help us to find ways to overcome them.
(This is joint work with Jack Keeler, Antoine Gaillard, Jack Lawless, Joao Fontana, Andrew Hazel and Anne Juel.)
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Elsen Tjhung (Durham)
Splay-bend phases in 2d liquid crystals: long-range order and beam-splitter
The usual nematic phase in liquid crystals is formed by rod-shaped molecules. However many liquid crystalline-forming molecules in nature are not straight. For instance, one can imagine banana-shaped or cone-shaped molecules, which can form various splay-bend phases. In this work, we predicted, theoretically, a new liquid crystalline phase, which we call splay-bend-infinity phase. This phase has an interesting optical property: it can split incoming light beam into two. This might have future technological applications such as 3d displays and augmented reality. From the fundamental point of view, this new liquid crsytalline phase also has a true long range order in 2d (in contrast to quasi-long range order in 2d nematics).
Reference: X. Ma and E. Tjhung, Banana- and pizza-slice-shaped mesogens give a new constrained O(n) ferromagnet universality class, Phys. Rev. E, 100, 012701 (2019)
MS22 Theory and modelling of liquid crystalline fluids
Ruben Tomlin (Imperial)
Instability and dripping of electrified liquid films
We utilise electric fields to control the interfacial dynamics of thin films of viscous dielectrics on inclined flat substrates. For certain choices of conductivities and permittivities, electric fields parallel to the flow direction may be used to stabilise a film interface. In particular, for films hanging from a tilted substrate, we are interested in ascertaining whether this physical mechanism can be utilised to suppress dripping, and if so, determine physical feasibility. Experimental work and analysis in the case of a non-electrified flow found the onset of dripping to be near to the absolute-convective transition in parameter space [Brun et al., Phys. Fluids 27, 084107, (2015)]. We investigate the influence of an external electric field on this result, utilising a hierarchy of reduced-order models to predict the spatial stability of the system, and perform direct numerical simulations (DNS) of the Navier--Stokes equations coupled to the relevant electrical equations. The latter is performed on a very long computational domain with inflow/outflow boundary conditions to mimic an experimental set-up. For the electrified flow, we find that dripping due to the instability of individual travelling pulses is close to the threshold of absolute instability. However, we also find non-transient dripping for electric field strengths beyond this threshold due to the merging of pulses (we expect this to be due to the nonlocality of the electric field effect). Nevertheless, the absolute instability threshold provides an order-of-magnitude estimate for the electric field strength required to suppress dripping completely, a useful prediction of which is not forthcoming from a temporal stability analysis alone. Although these linear and nonlinear phenomena do not coincide precisely, the computationally expensive DNS can be substantially more targeted with the predictions of absolute instability from the reduced-order models. We expect this approach to be valuable for other multi-physics problems.
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Lewis Topley (Birmingham)
Finite W-algebras and the orbit method
In a recent work Losev constructed a version of the orbit method for semisimple Lie algebras: this is a map from the set of coajoint orbits to the primitive spectrum of the enveloping algebra. For classical Lie algebras the map is known to be injective and Losev conjectured that the image should consist of primitive ideals obtained from one dimensional representations of W-algebras via Skryabin's equivalence. In this talk I will explain my proof of this conjecture. One of the key steps is the use of Dirac reduction to obtain new presentations of the semiclassical limits of finite W-algebras.
BMC01 Algebra and Representation Theory
Miloslav Torda (Liverpool)
Geometry of the n-torus stochastic trust region method for materials discovery.
Crystal structure prediction (CSP) is a widely used approach in materials science to guide new materials discovery. Current methods of CSP involve global searches in the space crystals. Given pressure-temperature conditions and a crystallographic space group the goal is to find a structure that minimizes lattice energy often represented as force fields that give rise to complicated energy landscapes with many basins of attraction. We propose a novel global search method based on stochastic trust region approach, as a variant of the natural gradient descent that exploits the structure of crystallographic space groups. We propose a parametric family of probability distributions defined on a n-dimensional torus, called extended multivariate von Mises distribution, that has a structure of a statistical manifold, and perform the search on this functional space. To further improve the performance of the stochastic trust region method we define an adaptive selection quantile and show a connection to the simulated annealing approach where the stochastic trust region method could be regarded as running multiple simulated annealing schedules in parallel. We examine the geometry of the stochastic trust region method with the adaptive selection quantile and show that it is a form of entropic proximal algorithm where the kernel in the infimal convolution is given by Kullback-Leibler divergence where in fact a minimization of Kullback-Leibler divergence between two statistical manifolds is performed in parallel. As a proof of concept we apply our method to the problem of finding densest packings of convex polygons in 2-dimensional space groups that is closely related to the crystal structure prediction problem.
CT05 Optimisation
Alvaro Torras Casas (Cardiff)
PerMaViss: distributed Persistent Homology with increased insight
Topological Data Analysis is mainly concerned about the shape of data. The main tool is "Persistent Homology" (PH), which detects cycles, holes and connected components in the data. This has been applied successfully on areas such as protein compressibility, pattern detection, machine learning techniques, and many more. However, persistent homology has some computational limitations; its algorithm is very expensive in terms of memory. In this talk, we outline a new method that allows us to join local computations and use them for computing PH. This approach comes from adapting a classical tool from topology; the "Mayer-Vietoris spectral sequence", to the setting of PH. Additionally, we will show that this technique leads to a more insight on PH than the classical methods. Finally, we present PerMaViss, a python3 library that implements these ideas.
Poster
Carlos Torres-Ulloa (Strathclyde)
Viscous froth model applied to the motion of three bubbles in a channel
The viscous froth model is used to study the rheology of a two-dimensional liquid-foam system comprised of three bubbles confined between two glass plates. The model balances the pressure difference across foam films with the surface tension acting along them coupled with curvature, converting any mismatch between the pressure forces and film curvature to film motion, which leads to viscous drag forces. The case studied here, i.e. the three-bubble case, interpolates between a so-called infinite staircase and the simple staircase/lens system. In the infinite staircase structure, the system does not undergo bubble neighbour exchange, i.e., topological transformations, for any imposed driving back pressure. Bubbles then flow out of the channel of transport in the same order in which they entered it. In contrast, in the simple single bubble staircase or so-called lens system, for higher imposed back pressures, topological transformations do occur. Steady state solutions are obtained for the three-bubble system for a range of bubble sizes and imposed back pressures. As the migration velocity of the system is increased quasi-statically from equilibrium, by applying an increasing imposed back pressure, systems undergo either topological transformations, reach saddle-node bifurcation points, or asymptote to an inherently stable structure which ceases to change as the back pressure is further increased.
MS24 The mathematics of gas-liquid foams
Gareth Tracey (Oxford)
Invariable generation and the Chebotarev invariant of a finite group
Given a finite group $X$, a classical approach to proving that $X$ is the Galois group of a Galois extension $K/\mathbb{Q}$ can be described roughly as follows:\\
(1) prove that $\Gal(K/\mathbb{Q})$ is contained in $X$ by using known properties of the extension (for example, the Galois group of an irreducible polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ embeds into the symmetric group $\Sym(n)$);\\
(2) try to prove that $X = \Gal(K/\mathbb{Q})$ by computing the Frobenius automorphisms modulo successive primes, which gives conjugacy classes in $\Gal(K/\mathbb{Q})$, and hence in $X$. If these conjugacy classes can only occur in the case $\Gal(K/\mathbb{Q})=X$, then we are done. The \emph{Chebotarev invariant} of $X$ can roughly be described as the efficiency of this `algorithm'.
In this talk we will define the Chebotarev invariant precisely, and describe some new results concerning its asymptotic behaviour.
BMC09 Groups
Matt Tranter (Nottingham Trent)
Initial-value problem for the Boussinesq-Klein-Gordon and coupled Boussinesq equations
In this talk I will discuss the so-called “zero-mass contradiction” that can emerge when constructing weakly-nonlinear solutions to Boussinesq-type equations for periodic functions on a finite interval. Firstly I will overview the recent results for the Boussinesq-Klein-Gordon equation, where a solution was constructed that takes account of this issue and numerical results will justify this approach. I will then consider the coupled Boussinesq equations. Solutions will be constructed for the cases when the characteristic speeds in the equations are close and when they differ significantly. Numerical results for solitary and cnoidal wave initial conditions will be presented and the energy in the system will be examined. These equations can model the propagation of long nonlinear longitudinal bulk strain waves in a two-layered elastic waveguide with a soft bonding between the layers, where the material in the layers determines the type of equation. This is joint work with Karima Khusnutdinova.
CT12 Waves
Dmitri Tseluiko (Loughborough)
Deformation and dewetting of liquid films under gas jets
We study the deformation and dewetting of liquid films under impinging gas jets using experimental, analytical and numerical techniques. We first derive a reduced-order model (a thin-film equation) based on the long-wave assumption and on appropriate decoupling of the gas problem from that for the liquid. To model wettability we include a disjoining pressure. The model not only provides insight into relevant flow regimes, but also is used to guide the more computationally expensive direct numerical simulations (DNS) of the full governing equations, performed using two different approaches, the Computational Fluid Dynamics package in COMSOL and the volume-of-fluid Gerris package. We find surprisingly that the model produces good agreement with DNS even for flow conditions that are well beyond its theoretical range of validity, and we analyse under which conditions dewetting is dominated by the gas jet and/or the receding contact line motion. We additionally compare the computational results with experiments and find good agreement.
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Daniel Tubbenhauer (Bonn)
2-representation theory of Soergel bimodules
This talk is a friendly introduction to 2-representation theory, with the emphasis on example such as 2-representations of Soergel bimodules (the categorical analog of representations of Hecke algebras).
BMC01 Algebra and Representation Theory
Vladimir Turetsky (Ort Braude, Israel)
Linear system identification by feedback control
A problem of restoration of time-dependent coefficients of a linear ordinary differential equation, based on the noised output measurements, is considered. This problem is ill-posed and needs a regularized solution. In the proposed algorithm, the coefficient vector is approximated by an optimal feedback control time realization in the auxiliary tracking problem. Two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields a linear-quadratic optimal control problem having a known explicit solution. Note that these derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter needs a numerical solution of a corresponding Bellman equation. Simulation results show that a bilinear model provides more accurate coefficients estimates.
CT05 Optimisation
Michel van Garrel (Birmingham)
Stable maps to Looijenga pairs
Start with a rational surface Y with a decomposition of its anticanonical divisor into at least 2 smooth nef components. We associate 5 enumerative theories to such Looijenga pairs: 1) all genus stable log maps with maximal tangency to each boundary component; 2) the all genus open Gromov-Witten theory of a toric Calabi-Yau threefold associated to the Looijenga pair; 3) genus 0 stable maps to the local Calabi-Yau surface obtained by twisting Y by the sum of the line bundles dual to the components of the boundary; 4) the Donaldson-Thomas theory of a symmetric quiver specified by the Looijenga pair and 5) BPS invariants associated to the various curve counting theories. In joint work with Pierrick Bousseau and Andrea Brini, we provide closed-form solutions to essentially all of these invariants and show that the theories are equivalent. I will focus on the relationship between 1) and 2) and present how the well-understood BPS theory of 2) leads to refinements of sheaf counting invariants of local Calabi-Yau fourfolds.
BMC07 Algebraic Geometry
Raul van Loon (Swansea)
Personalising circulatory models in pregnancy
The role of Doppler waveforms in the uterine arteries for clinical decision making has been explored by many research groups over the years. Various indices have been developed based on these waveforms that are used as indicators for poor placental development. However, these indices are still not well understood. Furthermore, the effect of the larger maternal circulation on the local utero-ovarian waveforms is not clear.
A computational framework will be presented that can be used to create personalized models of the maternal cardiovascular system. The basis of this framework is a one-dimensional, closed-loop model of the maternal circulation including all the main arterial and venous vessels with particular emphasis on the pregnant utero-ovarian circulation. To personalise this model an automated optimisation algorithm was developed that integrates non-invasive data, i.e. heart rate, cardiac output, peripheral resistance, systolic & diastolic blood pressure, aorta size, pulse wave velocity and uterine Doppler waveforms. The resulting models provide us with pressure and flow waveforms in both the larger maternal vessels and the smaller utero-ovarian circulation for each patient and as a result, allows us to study their relation.
This framework was used to create personalised models of 20 pregnant women, 10 of whom developed pre-eclampsia throughout their pregnancy. The predicted uterine waveforms were compared to the measured ones. Furthermore, initial results suggest that the cross-sectional areas of the larger maternal vessels could be an early predictor of pre-eclampsia. Further research is required to consolidate these findings.
MS27 Mathematical and Computational Modelling of Blood Flow
James Van Yperen (Sussex)
Modelling COVID-19 for Sussex healthcare demand and capacity
At the beginning of the COVID-19 crisis in the UK, Sussex Health and Care Partnership developed a Gold Command structure which posed modelling questions of strategic operational significance to the Local Health Resilience Panel covering East Sussex, West Sussex, and Brighton & Hove. As a result, the Sussex Modelling Cell (SMC) was established which includes leaders from Public Health Intelligence and academics from the University of Sussex. In this talk I will present the SMC healthcare demand and capacity model, how it has evolved over the epidemic, and how we will make it available for other local regions to use across the country - our web-based translational toolkit, Halogen.
MS28 Covid-19 Modelling
Alina Vdovina (Newcastle)
Buildings, C*-algebras and new higher-dimensional analogues of the Thompson groups
We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their $K$-theory. The underlying building structure allows explicit computation of the $K$-theory. We will also present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the $K$-theory ofC*-algebras gives new invariants to recognize non-isomorphic groups.
BMC04 Operator Algebras
Christian Vergara (Milano)
Modeling blood flow: numerical methods and clinical applications
In this talk we report some examples of cardiovascular applications in which computational models are used to properly quantify outputs of clinical interest. In particular, we focus on blood flow modelling, describing different mathematical and numerical models, such as Large Eddy Simulations for transition to turbulence and Fluid-Structure interaction problems. We then present examples of possible clinical applications made in collaboration with hospitals.
MS27 Mathematical and Computational Modelling of Blood Flow
Mats Vermeeren (Leeds)
A Lagrangian perspective on integrability
We present a Lagrangian perspective on integrable systems. While in mechanics the Hamiltonian and Lagrangian formulations are largely interchangeable, the theory of integrable systems is typically formulated in the Hamiltonian language only. The notion of Liouville-Arnold integrability puts the physical Hamilton function and the Hamilton functions describing its symmetries on the same footing. Hence one can think of solutions not just as a function of time, but of a "multi-time" containing a parameter for each symmetry flow. Symmetries also have an important place in the Lagrangian picture, in particular in Noether's theorems. However, these treat the symmetries as fundamentally different from the Lagrange function. The Lagrange function and its symmetries can be put on the same footing by using a variational principle in multi-time, known as the "pluri-Lagrangian" principle or the "Lagrangian multi-form" principle. The reach of this theory extends far beyond mechanics, capturing also integrable partial differential equations and lattice equations.
Poster
Nelly Villamizar (Swansea)
Supersmooth splines and ideals of mixed powers of linear forms
The study of multivariate piecewise polynomial functions (or splines) on polyhedral complexes is important in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design.
In the talk we will address various challenges arising in the study of splines with enhanced (super-)smoothness conditions at the vertices or mixed smoothness across the interior the faces of the partition. Such super-smoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Understanding these splines involve the analysis of ideals generated by products of powers of linear forms in several variables. We will present some of the algebraic tools used to study these spaces, as well as some open questions, and various examples to illustrate the approach. This is a joint work with D. Toshniwal.
MS07 Applied Algebra and Geometry
Laura Wadkin (Newcastle)
Using mathematics to understand stem cell pluripotency regulation
Human pluripotent stem cells, hPSCs, hold great promise for developments in regenerative medicine and are at the forefront of modern biological research due to their ability to differentiate into any type of human adult cell (the pluripotency property) and their potential to self-renew indefinitely through repeated divisions. As part of an interdisciplinary team at Newcastle University, I am working to optimise experiments by modelling the behaviour of hPSC colonies using a combination of agent-based and stochastic techniques. Having already modelled colony proliferation to optimise colony clonality experimentally, we now focus on modelling the defining property of stem cells, pluripotency. We use a stochastic logistic equation containing both multiplicative and additive noise to capture the temporal shifts in pluripotency over time as seen experimentally. Analysis of the experimental data also shows pluripotency is anti-persistent, revealing self-regulation in the system. Representative models of individual cell pluripotency further our understanding of the inherent biological behaviours of stem cells, provide the basis for modelling pluripotency spatially across colonies and assist in experimental optimisation.
CT 16 Mathematical Biology-3
Nathalie Wahl (Copenhagen)
Strings in manifolds
Chas and Sullivan defined 20 years ago a “topological string interaction” on manifolds and asked what these interactions could tell us about the manifold they live in. I’ll give an introduction to this rich world, now known as string topology, that has since spread its tentacles to algebra, geometry or even geometric group theory.
BMC Morning Speaker
William Waites (LSHTM)
A compositional account of immune response, testing and virus transmission
Transmission models in infectious disease traditionally subdivide the population into compartments, and individuals are said to move between these compartments as they become infected, their disease progresses in various ways and they ultimately recover or succumb. This abstraction is useful and productive, but it does not readily admit study of how the underlying interaction of individuals’ immune responses to the virus influences population-level dynamics. This influence is important because it governs both infectiousness and time-varying diagnostic test response used to trigger interventions. We demonstrate how the interaction of immune response and virus population within an individual can be described within a rule-based, stochastic graph rewriting formalism. This formulation reproduces empirically observed diagnostic test response. We further demonstrate how such a model can be extended to transmission and how it suggests at a partial biological basis for superspreading phenomena. Finally, we show that the abstraction of compartments can be recovered simply by an appropriate choice of graph observables.
MS28 Covid-19 Modelling
Alan Walker (University of the West of Scotland)
Mathématiques sans Frontières
Mathématiques sans Frontières is an international mathematical competition for school children, centrally organised by the Académie de Strasbourg. It aims to promote mathematics, teamwork, and the practice of a foreign language. Presently, the competition has a good take-up in Scotland, with academics from the University of the West of Scotland organising the UK-side of the competition.
I’d like to expand interest in the competition. If you’d like to help promote the competition in Scotland, get involved with translation and/or marking, or if you’d like to organise the competition in other parts of the UK, or in the Republic of Ireland, then please do get in touch.
Outreach Video
Simon Walker-Samuel (UCL)
Combining optical imaging of cleared tissue with mathematical modelling and in vivo imaging to predict drug delivery and therapeutic response
Understanding how drugs are delivered to diseased tissue, and their subsequent spatial and temporal distribution, is a key factor in the development of effective, targeted cancer therapies. Preclinical tools to better understand drug delivery are urgently required, which incorporate the inherent variability and heterogeneity between tumour types and deposits, and even within individual tumours[1]. However, few (if any) experimental techniques exist that can quantify drug delivery across whole tumour samples purely through experimental imaging. To meet this need, we have developed the REANIMATE (REAlistic Numerical Image-based Modelling of biologicAl Tissue substratEs) framework[3], which integrates optical imaging of intact biological tissue with computational modelling. Specifically, REANIMATE enables the microstructure of these large samples to be virtually reconstructed in 3D, on the scale of microns. These resultant data act as substrates for our mathematical model which is parametrised and validated against in vivo ASL-MRI perfusion data, thereby enabling physiological simulations of fluid delivery through the vasculature and into the surrounding tissue[3]. REANIMATE was applied to imaging data from two murine models of colorectal cancer (LS147T and SW1222) to: 1) simulate steady-state fluid dynamics (such as intravascular and interstitial fluid pressure (IFP)), 2) uptake of the MRI contrast agent Gd-DTPA, and 3) uptake and response to vascular-targeting treatment (Oxi4503). REANIMATE predictions were found to be consistent with the magnitude and spatial heterogeneity of in vivo measurements, both in steady-state (blood flow, IFP) and transient (drug delivery) models[3]. Simulations predicted that, whilst the traditionally elevated IFP in the tumour core[4] can occur, vascular spatial heterogeneity can also induce spatially heterogeneous IFP[3]. Lastly, loss of vessels as a result of administration of Oxi4503 resulted in a subtle spatial pattern of perfusion loss in significant tissue volumes that is tumour-type dependent. [1] A. Alizadeh et al., Nature Medicine, Vol. 21(8), 846-853 (2015) [2] H. Rieger et al. WIREs Syst Biol Med, 7, 113-129 (2015) [3] A. d’Esposito et al., Nat. Bio. Eng., 2, 773-787 (2018) [4] L. Baxter et al., Microvascular Research, 37(1), 77-104 (1989)
MS25 Multiscale modelling, simulations, and experiments. Interdisciplinary challenges and applications to real-world biophysical systems
Josh Walton (Strathclyde)
Spontaneous flow transitions in active nematic liquid crystals
Active nematic liquid crystals are fluids in which continuous internal energy generation, such as in bacterial suspensions and microtubule-forming suspensions, allows for spontaneous flow generation. In such fluids, the flow-generating agent is usually anisotropic (defined by, for instance, the long axis of the bacterium or microtubule) with such symmetry giving rise to a liquid crystalline-like phase. Internally driven flows can lead to interesting effects such as self-organisation and non-equilibrium defect configurations. We consider the director orientation and flow of an active nematic liquid crystal confined between two parallel glass plates. Our theoretical analysis is based on extended Ericksen- Leslie equations such that the stress tensor comprises of the usual liquid crystal viscous stress and an additional active stress term which accounts for the activity of the fluid. We assume planar anchoring of the director on the plates and no-slip conditions for the ow velocity.
Decoupling the director angle and fluid velocity in the Ericksen-Leslie equations allows us to analytically calculate critical activity strengths at which the active fluid will spontaneously transition from a non-flow state to a flow state. This phenomena is analogous to a Freedericksz transition in inactive nematic liquid crystals, where the director begins to re-orient when an applied electric field exceeds a critical threshold value. Our analytic results agree very well with the corresponding numerical calculations of the full non-linear model. An asymptotic approach for large activity allows us to understand the behaviour of these solutions; specifically, we predict the possible values of the director orientation angle in the bulk of the channel. The steady state Ericksen-Leslie equations are then solved numerically to obtain non-trivial solutions for varying activity. The stability of the various non-trivial solutions is also determined. We find that changes in the activity magnitude lead to interesting effects such as flow alignment.
MS22 Theory and modelling of liquid crystalline fluids
Yixuan Wang (Pittsburgh)
Strong solution for compressible liquid crystal system with random force
We study the three-dimensional compressible Navier-Stokes equations coupled with the $Q$-tensor equation perturbed by a multiplicative stochastic force, which describes the motion of nematic liquid crystal flows. The local existence and uniqueness of strong pathwise solution up to a positive stopping time is established where ``strong" is in both PDE and probability sense. The proof relies on the Galerkin approximation scheme, stochastic compactness, identification of the limit, uniqueness and a cutting-off argument. In the stochastic setting, we develop an extra layer approximation to overcome the difficulty arising from the stochastic integral while constructing the approximate solution. Due to the complex structure of the coupled system, the estimates of the high-order items are also the challenging part in the article.
MS21 Mathematical and Physical Challenges in Anisotropic Soft Matter
Chen Wang (Exeter)
Instability in shallow-water magnetohydrodynamics with magnetic shear
In this study, we consider the linear instability of shallow water shear flow with a magnetic field. Shallow water MHD was first proposed by Gilman (2000, \textit{J. Astrophys.} 544, L79) as a reduced system for studying certain astrophysical flows, particularly those in the solar tachocline. The basic magnetic field is parallel to the basic flow velocity and has a shear in the cross-stream direction. The flow admits critical levels where the neutral modes become singular at the locations where the phase velocity relative to the basic flow equals the Alfv{\'e}n wave velocity. Previous studies have shown that a constant magnetic field generally has a stabilizing effect (Mak \textit{et al.} 2016, \textit{J.Fluid Mech.} 788, 767). Here we show that in general, this is also the case when the magnetic field varies with space. However, in situations where two critical levels are close to each other, the magnetic field could generate a new instability. The second-order spatial derivative of the magnetic field plays the crucial role, in a mechanism similar to the critical-layer vorticity gradient of hydrodynamic instability.
CT11 Magnetohydrodynamics
Danyang Wang (Glasgow)
The energetics of flow in a flexible channel with nonlinear fluid-beam model
We consider flow along a finite-length collapsible channel, where one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to a uniform external pressure. A modified constitutive law is used to ensure that the elastic beam model is both geometrically and materially nonlinear, while remaining energetically conservative. A parabolic inlet flow with constant flux is driven through the channel, with a prescribed fluid pressure at the downstream end. We apply the finite element method with adaptive meshing to numerically solve the fully nonlinear steady and unsteady systems [1]. In line with previous studies, we show that across the parameters investigated the system always has at least one static solution; in addition there is a narrow region of the parameter space where the system exhibits two stable static configurations simultaneously, consisting of an upper branch (where the steady beam is almost entirely inflated) and a lower branch (where the steady beam is collapsed) [2]. These two branches of static solutions are connected in the parameter space by an unstable intermediate branch, joined at a pair of limit point bifurcations. We show that both the upper and lower static configurations can each become unstable to self-excited oscillations, whereas only lower branch instabilities had previously been noted in this fluid-beam system. We consider a detailed study along two slices through the parameter space, showing that for fixed beam elasticity and increasing Reynolds number the upper branch is first to become unstable to oscillations via a supercritcal Hopf bifurcation; these oscillations onset close to, but just outside, the region of parameter space with multiple static states. As the Reynolds number increases further the unstable branch enters the region with multiple steady states and eventually stabilises again very close to the limit point of the upper branch static solutions. As the Reynolds number increases further the lower branch of static solutions becomes unstable to oscillations via a supercritical Hopf bifurcation, again in a region of parameter space close to, but just outside, the region with multiple static states. Furthermore, our new formulation allows us to self-consistently calculate a detailed energy budget over a period of fully developed oscillation. We show that, for both the upper and lower branch instabilities, the oscillation requires an increase in the work done by the upstream pressure to overcome the corresponding increase in dissipation in the oscillatory flow.
[1] Luo, X. Y., Cai, Z. X., Li, W. G., & Pedley, T. J. (2008). The cascade structure of linear instability in collapsible channel flows. Journal of Fluid Mechanics, 600, 45-76.
[2] Stewart, P. S. (2017). Instabilities in flexible channel flow with large external pressure. Journal of Fluid Mechanics, 825, 922-960.
CT17 Solid Mechanics
Ben Ward (York)
P-adic Diophantine Approximation on Curves
In a recent paper Beresnevich, Lee, Vaughan and Velani found a lower bound of the Hausdorff dimension for the set of simultaneously approximable points over a manifold in n-dimensional real space. This poster is motivated by a similar technique that has been applied to obtain a bound for the set of p-adic weighted approximable points over a n-dimensional p-adic curve. To give such bound certain restrictions are applied to both the approximation functions and the properties of the curve, however, these are reduced as much as possible by using properties of the p-adic norm.
Poster
Joern Warnecke (Max-Planck-Institut)
Modelling solar and stellar activity driven by turbulent dynamo effects
The magnetic field in the Sun undergoes a cyclic modulation with a reversal typically every 11 years due to a dynamo operating under the surface. Other solar-like stars with outer convective envelopes shows cyclic modulation of their magnetic activity, the level and cycle period being related to their rotation rate. This is suggestive of a common dynamo mechanism.
Here we present results of 3D MHD convective dynamo simulations of slowly and rapidly rotating solar-type stars, where the interplay between convection and rotation self-consistently drives a large-scale magnetic field. With the help of the test-field method, we are able to measure the turbulent transport coefficients in these simulations and therefore get insights about the dynamo mechanism operating in them. It allows us to explain the weak dependency of the cycle period found in the moderate rotation regime using a Parker dynamo wave operating in our simulations. Furthermore, we find that the alpha effect becomes highly anisotropic for high rotation rates, which can explain the high degree of non-axisymmetry of magnetic field in observations and models of rapid rotating stars.
Overall, the turbulent contributions to the electromagnetic force play an important role for dynamics and evolution of the large-scale magnetic field in all of our simulations. Stars spinning faster than the Sun are expected to also produce larger amounts of magnetic helicity at their surfaces. On the Sun, magnetic helicity is essential for the release of energy leading to the eruption of plasma via coronal mass ejection and it is thought to play an important role in the heating process of the coronal plasma. Using MHD simulations of solar coronae we find a power law relation between the surface magnetic helicity and the temperature and activity of these coronae, suggesting an important role of magnetic helicity production in understanding rotational dependence of stellar activity.
MS15 Recent Developments in Magnetohydrodynamics and Dynamo Theory
Edwin Watson-Miller (Bath)
Discrete and Continuous Snaking Bifurcations
There is a great deal of interest in nonlinear differential and difference equations that exhibit homoclinic snaking, in which solution curves of localised patterns 'snake' back and forth across a bifurcation diagram in a narrow region of parameter space. In this poster, we present two approaches for extending our understanding of snaking bifurcations in bistable nonlinear systems: (i) numerical techniques to understand limiting behaviours in solution structures, especially in discrete cases (lattices of coupled cells) in two and higher dimensions; and (ii) the use of exponential asymptotics to characterise the multiple-scales structure of the solutions and analytically describe the snaking phenomenon.
Poster
Denis Weaire (Dublin)
Kelvin wakes up in Largs: progress since then
Towards the end of a glittering career in physics and engineering, Sir William Thompson retained an interest in fundamental questions: in particular, what medium carries a light wave? He awoke one morning in his country home in Largs, with a sudden inspiration: that the medium was a kind of foam, whose energy he set out to minimise. This was the first such analysis, apart from the statement of general principles by Plateau. The study of foam structure has been continued for more than a century: we review some recent examples, as well as the broader cultural aftermath of the “Kelvin Problem”.
Acknowledgements: European Space Agency: "Soft Matter Dynamics" projects, contracts 4000115113 and 4000129502.
MS24 The mathematics of gas-liquid foams
Cat Wedderburn (Edinburgh)
Burn, baby, burn: Mathematical firefighting to reduce potential disease spread
The Firefighter game offers a simple, discrete time model for the spread of a perfectly infectious disease and the effect of vaccination. A fire breaks out on a graph at time 0 on a set F of f vertices. At most d non-burning vertices are then defended and can not burn in future. Vertices, once either burning or defended remain so for the rest of the game. At each subsequent time step, the fire spreads deterministically to all neighbouring
undefended vertices and then at most d more vertices can be defended. The game ends when the fire can spread no further. Determining whether k vertices can be saved is NP-complete. I focus on finding maximal minimal damage (mmd) graphs - graphs which have the least burning if the fire starts in the worst place and the defenders defend optimally.
I shall present some new and old results linking mmd graphs to optimal graphs for the Resistance Network Problem of finding graphs where all F-sets of vertices have limited neighbourhoods; a new framework for proving graphs are mmd and a new algorithm for optimal defense of a graph under certain conditions. I shall then present some results using the discharge method on the edge defence version of the Firefighter game and compare them to results in the original game.
CT04 Mathematical Biology-1
Arkady Wey (Oxford)
A size-structured filtration model
Filters are used in industry to separate impurities and harmful particulates from solution, with applications ranging from high-volume industrial emissions abatement to the processing of blood samples. W.L. Gore and Associates supply businesses with particle filtration products. These are constructed from fibres that form a complex network of pores, within which particles can become trapped as the filtration process progresses. However, as the pores capture contaminants, they become blocked. The filters gradually clog, preventing further particle removal. Clogging greatly increases downtime and running costs of filtration processes, leading to decreased overall productivity. It also often leads to filter disposal, to the detriment of the environment.
We will present a size-structured mathematical model that predicts contaminant removal and clogging. In our model, we track the concentration distribution of particles and pores. The model comprises two coupled integro-partial-differential equations for the particles and pores, along with Darcy’s equation for the flow. We solve the model numerically in a number of scenarios, and compare and contrast the results.
CT08 Industrial fluids
Andy White (Heriot-Watt)
Mathematical Modelling Tools for Red Squirrel Conservation
The invasive North American grey squirrel has replaced the native Eurasian red squirrel in most of England, Wales and parts of Scotland and Ireland. Grey squirrels are generally better competitors for resources and additionally carry a disease, squirrelpox, which is fatal to red squirrels. We modify a general deterministic model framework that represents competition and disease interactions between red and grey squirrels to include explicit spatial and stochastic processes. Current efforts to aid the conservation red squirrels in Scotland include the designation and management of stronghold forests that are intended to provide refuge for red squirrels against the incursion of grey squirrels. We use the mathematical model to assess the population viability of red squirrels in these strongholds and discuss how the results can be used to inform a forthcoming review of stronghold management policy by Scottish Forestry.
MS13 Mathematical challenges in spatial ecology
Neshan Wickramasekera (Cambridge)
Allen–Cahn equation and the existence of prescribed-mean-curvature hypersurfaces in Riemannian manifolds
An n-dimensional hypersurface of a Riemannian manifold has n principal curvatures (relative to a chosen unit normal vector field) at each of its points. The sum of the principal curvatures is the scalar mean curvature, which is a real valued function on the hypersurface. A basic question in Riemannian geometry is whether, given a real valued function g on the ambient manifold, there exists a boundaryless hypersurface, together with a choice of unit normal, such that the scalar mean curvature of the hypersurface (relative to the unit normal) is given by g at every point. The case g = 0 corresponds to extensively-studied minimal hypersurfaces. The lecture will discuss progress on this question for closed ambient manifolds, focusing on a recently developed PDE theoretic approach (joint work with Costante Bellettini at UCL). This method utilises the elliptic and parabolic Allen–Cahn equations on the ambient space. It brings to bear on the question certain elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic PDE theory—principles that serve as a conceptually and technically simpler replacement for the Geometric Measure Theory machinery developed four decades ago in the pioneering work of Almgren and Pitts to prove existence of a minimal hypersurface. A key outcome ofthe PDE method is an affirmative answer to the above question, in all dimensions n ≥ 2, when g is any non-negative (or non-positive) Lipschitz function: for such g, any closed Riemannian manifold contains a C 2 quasi-embedded, boundaryless, mean-curvature-g hypersurface (which, if n ≥ 7, may contain a closed singular set of Hausdorff dimension ≤ n − 7).
BMC Morning Speaker
Hanneke Wiersema (Kings College London)
On a BSD-type formula for L-values of Artin twists of elliptic curves
In this talk we will discuss the possible existence of a BSD-type formula for L-functions of elliptic curves twisted by Artin representations. After outlining some expected properties of these L-functions, we will present arithmetic applications and some explicit examples. This is joint work with Vladimir Dokchitser and Robert Evans.
BMC02 Number Theory
Paul Wilcox (Bristol)
Turning Ultrasonic Array Data into Structural Integrity Information
Ultrasound is a key modality for the Non-Destructive Evaluation (NDE) of engineering assets ranging from aircraft and wind turbines to nuclear power stations and pipelines. Modern ultrasonic arrays allow the acquisition of high-fidelity digital data in unprecedented volumes. A typical frame of raw ultrasonic array data contains of the order of 1000 time-trace associated with every possible combination of transmit-receive elements in the array, and each time-trace contains of the order of 1000 points. The goal of NDE is to turn this raw data into useful information about the integrity of a structure; this is implicitly an inverse problem, even if it is not always formally recognised as one. In this talk, the various ways in which the problem has been approached since ultrasonic arrays first appeared in the 1970s will be briefly reviewed. Some current avenues of research will then be discussed, including the many variants of classical delay-and-sum imaging, adaptive imaging, multi-view imaging, and reversible imaging. Methods of quantitative characterisation of defects in engineering structures will also be considered. These include scattering matrix extraction and analysis, which enables sizing below the classical diffraction limit, and machine learning. Finally, some perspectives on future directions for both data acquisition and processing will be given.
MS10 Ultrasonic Waves
Christophe Wilk (Manchester)
Wetting and dewetting dynamics of a thin liquid film spreading on an immiscible liquid surface
Macroscopic thin liquid films are present in many processes ranging from coating flow technology to biological systems such as fluid lining in the pulmonary airways [1] or in the formation of tear film. Thin liquid films also provide a model mesoscopic system to study directly the effects of fundamental scientific issues such as intermolecular forces. In this study, experiments on the spreading of surfactant-laden immiscible droplets over liquid substrates demonstrated a striking and unusual dewetting instability [2]. As the film spreads over the liquid substrate due to Marangoni forces, it reaches, at least locally, a critical thickness under which intermolecular forces becomes important [3]. Then, a van der Waals induced thinning process leads to the appearance and growth of holes (regions of minimal film thickness) at the surface of the spreading liquid film while its leading edge is still expanding. After the end of the spreading phase, these apparent holes continue to form and grow. The interplay between Marangoni and intermolecular forces enables the system to wet and apparently dewet the substrate at the same time, which seems counter-intuitive. We have studied the system both experimentally and theoretically. Analysis of the experimental data shows that the spreading is driven by a Marangoni flow and that the dewetting velocity (the rate at which the holes grow) is constant across the film interface. The pattern of holes formed in the process was analysed using Minkowski functionals which showed that holes nucleate following random processes rather than spinodal dewetting, in agreement with past literature on solid substrate configurations [4]. Our model, based on the lubrication approximation [5], attempts to explain the key physical mechanisms responsible for the dewetting phenomena observed experimentally. Linear stability analysis of a system consisting of two superposed layers, a thin film on a thick substrate, of immiscible liquids resting on a solid substrate is performed [6]. The model includes intermolecular forces (with a long-range attractive component and a short-range repulsive one), capillary forces, and insoluble surfactant in the film. We investigate the most unstable modes to identify the key parameters that can lead to the dewetting phenomena observed experimentally. We discuss our model predictions in comparison with our experimental findings.
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Bibliography:
[1] Jensen and Grotberg, J. Fluid Mech. 240, 259 (1992).
[2] Shahidzadeh et al., Phys. Rev. E. 64, 021911 (2001).
[3] Sharma and Bandyopadhyay, J.Chem. Phys. 125, 054711 (2006).
[4] Becker et al., Nat. Mater. 2, 59 (2003).
[5] Craster and Matar, Langmuir 23, 5, 2588-2601 (2007).
[6] Karapetsas et al., Phys. Fluids 23, 122106 (2011).
CT08 Industrial fluids
Gareth Willetts (Exeter)
Efficient calculation of the H2 Norm via the polynomial Diophantine equation
The H2 norm is a widely used metric for measuring system performance. An example is its use in mechanical networks, such as suspension systems, where the performance can be quantified by the H2 norm from the tyre displacement to the chassis displacement, and a well performing suspension system will seek to minimise this H2 norm. This project derives a new and explicit method for computing the H2 norm of a single input single output system directly from the coefficients in its transfer function. The method uses Cauchy’s residue theorem to prove that the H2 norm can be obtained from the solution to a polynomial Diophantine equation. We also present a new efficient fraction-free algorithm for solving such polynomial Diophantine equations, inspired by the extended Euclidean algorithm and the concept of polynomial subresultant and subremainder sequences. The algorithm also returns the so-called Hurwitz determinants of the system which allows for a stability test to be performed with no added computational cost. This method has provided promising results, calculating H2 norms faster than current implementations existing in both MATLAB and Maple, and can be implemented numerically and symbolically.
Poster
Stephen Wilson (Strathclyde)
Competitive Evaporation of Multiple Thin Droplets
The evaporation of one or more sessile droplets is a very exciting and dynamic area of interdisciplinary study in fluid mechanics, with worldwide activity across many different subject disciplines, including physics, chemistry, mathematics, biology and engineering. However, while the vast majority of the previous work has concerned a single droplet, in practice, most droplets do not occur in isolation, and so interactions between droplets are of great practical and scientific interest. In particular, the critical difference between the evaporation of single and of multiple droplets is the occurrence of the so-called “shielding effect”, namely that the presence of other evaporating droplets increases the local vapour concentration in the atmosphere, and so each droplet evaporates more slowly than it would in isolation. In order to better understand this phenomenon, in the present talk we investigate two different but related scenarios, namely the evaporation of multiple thin droplets in three dimensions and the evaporation of one or two thin droplets in two dimensions. In particular, in both of these scenarios we are able to quantify the shielding effect that the droplets have on each other, and determine how it extends the lifetimes of the droplets. We also highlight fundamental differences between the two-dimensional and the three-dimensional droplet evaporation problems.
This is joint work with Alexander W. Wray, Brian R. Duffy, Feargus G. H. Schofield and David Prichard (also at the University of Strathclyde).
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Andrew Wilson (Glasgow)
maths inside Photo Competition
From Barra to Banff, Arran to Aberdeen, Kirkwall to Kilmarnock, and Skye to Scourie, the maths inside photo competition continues to succeed in engaging the entire nation in mathematics.
Launched back in 2018 to celebrate the Scottish Government initiative ‘Maths Week Scotland’, this transdisciplinary competition hosted by the University of Glasgow School of Mathematics & Statistics provides a framework for everyone to identify and reflect on the maths surrounding us: the maths inside.
In addition to the regular competition during Maths Week Scotland, maths inside came together with the Scottish Government and celebrated a Special Summer 2020 Edition to support families and communities during lockdown. This challenging 2020 has seen around 600 photo entries, and a further 470 screens attending the virtual prize ceremonies hosted by founder and director, Dr Andrew Wilson, who commented that:
"It is enormously rewarding. Year on year we are seeing all ages and backgrounds connect through shared positive mathematical experiences, exciting and challenging learning, phenomenal curiosity and artistic talent!"
To inspire entrants, and support parents, teachers and those out-of-school on their journeys of discovery, a suite of resources was also created in a collaborative project with undergraduate and postgraduate students during the first lockdown.
Enjoy the mathematical insights and beautiful images of the winners for yourself at mathsinside.com. Keep up-to-date with future editions on Twitter, Facebook, and Instagram. Experience the competition impact mathsinside.com/impact/
Outreach Video
Golo Wimmer (Imperial)
Upwind stabilised finite element schemes in a Poisson bracket framework
An important aspect for numerical weather prediction, particularly for long term simulations, is conservation of quantities such as mass and energy. One way to ensure the latter is to discretise the governing equations within a Poisson bracket framework, where the equations are inferred from the system’s Hamiltonian (i.e. the total amount of energy) and a Poisson bracket. In this presentation, we consider this framework for the compatible finite element method, which has recently been proposed as a discretisation method for numerical weather prediction and will be used in GungHo, the UK Met Office's next generation fluid dynamics component. The method allows the use of pseudo-uniform grids on the sphere that avoid the parallel computing issues associated with the latitude-longitude grid. It is also quite general, allowing for adaptive mesh refinement and higher-order discretisations.
We first review the incorporation of the Poisson bracket framework into the compatible finite element approach. We consider a Poisson bracket which, depending on the Hamiltonian, leads to the thermal shallow water equations or compressible Euler equations. Given shortcomings in the qualitative field development, we then introduce an extension to include upwind advection schemes for the different finite element spaces used in this approach, while still retaining energy conservation via the Hamiltonian setup.
MS14 Variational Methods in Geophysical Fluid Dynamics
Alexander Wray (Strathclyde)
Electrostatic control of thick coating flows on circular cylinders
We examine the behaviour of coating flows on the exterior of a rotating horizontal cylinder via the use of a novel long-wave model and direct numerical simulations (DNS). In contrast to classical thin-film lubrication theory-based models, the presented long-wave model is shown to allow the thickness of the film to be of the same order as the radius of the cylinder while still maintaining excellent agreement with DNS. The system is observed to exhibit a complex variety of behaviours owing to the interplay of rotation, viscosity, inertia, capillarity and gravity. The parameter space is explored comprehensively, with a variety of numerical and analytical techniques used to elucidate the physical and mathematical mechanisms for the observed transitions in behaviour.
In practical situations it is often desirable to exert control over the behaviour of such coating flows. By incorporating a concentric pair of electrodes leading to the production of a spatially varying potential, thereby exerting an electric stress at the liquid-gas interface, we show that the instabilities can be selectively augmented or suppressed. Comparison with simulations again show a high level of fidelity for the low-order model and show promise in the context of generating strong predictive capabilities for industrial applications.
MS19 Recent advances in multi-physics modelling and control of interfacial flows
Angela Wu (UCL)
Weinstein handlebodies for complements of smoother toric divisors
In this talk, I will introduce you to two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which come with a handle decomposition compatible with their symplectic structure. I will then show you an algorithm which produces the Weinstein handlebody diagram for the complement of a smoothed toric divisor in a "centered" toric 4-manifold. This is based on joint work with Acu, Capovilla-Searle, Gadbled, Marinković, Murphy, and Starkston.
BMC03 Topology
Jeremy Wu (Oxford)
Understanding the Landau Equation as a Gradient Flow
The Landau equation is an important PDE in kinetic theory modelling plasma particles in a gas. It can be derived as a limiting process from the famous Boltzmann equation. From the mathematical point of view, the Landau equation can be very challenging to study; many partial results require, for example, stochastic analysis as well as a delicate combination of kinetic and parabolic theory. The major open question is uniqueness in the physically relevant Coulomb case. I will present joint work with José Carrillo, Matias Delgadino, and Laurent Desvillettes where we cast the Landau equation as a generalized gradient flow from the optimal transportation perspective motivated by analogous results on the Boltzmann equation. A direct outcome of this is a numerical scheme for the Landau equation in the spirit of de Giorgi and Jordan, Kinderlehrer, and Otto. An extended area of investigation is to use the powerful gradient flow techniques to resolve some of the open problems and recover known results.
MS01 Challenges in Structure-Preserving Numerical Methods for PDEs
Runlian Xia (Glasgow)
Non-commutative Hilbert transforms and Cotlar-type identities
The Hilbert transform $H$ is a basic example of a Fourier multiplier. Its behaviour on Fourier series is the following:
$$\sum_{n\in \mathbb{Z}}a_n e^{inx} \longmapsto \sum_{n\in \mathbb{Z}}m(n)a_n e^{inx},$$ with $m(n)=-i\,{\rm sgn} (n)$. Riesz proved that $H$ is a bounded operator on $L_p(\mathbb{T})$ for all $1<p<\infty$.
We study Hilbert transform type Fourier multipliers on group algebras and their boundedness on corresponding non-commutative $L_p$-spaces. The pioneering work in this direction is due to Mei and Ricard, in which they prove $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity on von Neumann algebras. In this talk, we introduce a new form of Cotlar identities for groups that are not necessarily free products and study their validity for lattices of ${\rm SL}_2(\mathbb{C})$ and some other groups acting on trees.
Joint work with Adrián González and Javier Parcet.
BMC04 Operator Algebras
Vijay Kumar Yadav (Nirma . India)
Exponential synchronization of a class of fractional order complex chaotic systems and application through digital cryptography
In this article, exponential synchronization between a class of fractional-order chaotic systems has been studied. The exponential synchronization is analyzed by using exponential stability theorem for the fractional-order system, and the stability analysis has been done with the help of a new lemma, which is given for the Lyapunov function for the fractional-order system. The fractional-order Lorenz and Lu complex chaotic systems are considered to illustrate the exponential synchronization. The numerical simulations and graphical results are also presented to verify the effectiveness and reliability of exponential synchronization. The application in communication through digital cryptography is also discussed between the sender (transmitter) and receiver using the exponential synchronization. A well-secured key system of a message is obtained in a systematic and very simple way.
CT13 Dynamical Systems
Makoto Yamashita (Oslo)
Homology and K-theory of torsion free ample groupoids
The problem of connecting the integral homology to the K-groups of C*-algebra for ample groupoids was recently popularized by Matui. As an answer for this, we construct a spectral sequence starting from the groupoid homology which ends at the K-groups when the groupoid satisfies the Baum-Connes conjecture and has torsion free stabilizers. The construction crucially relies on the Meyer-Nest theory of triangulated structure on equivariant KK-categories. The same technique allows us to incorporate Putnam’s homology theory for Smale spaces in place of groupoid homology.
BMC04 Operator Algebras
Anthony Yeates (Durham)
Revisiting Taylor relaxation
Turbulent magnetic relaxation is an important candidate mechanism for coronal heating and some types of solar flare. By developing turbulence that reconnects the magnetic field throughout a large volume, magnetic fields can spontaneously self-organize into simpler lower-energy configurations. We are using resistive MHD simulations to probe this relaxation process, in particular to test whether a linear force-free equilibrium is reached. Such an end state would be predicted if one assumes the classic Taylor hypothesis: that the only constraints on the relaxation come from conservation of total magnetic flux and helicity. In fact, a linear force-free state is not reached in our simulations, despite the conservation of these total quantities. Instead, the end state is better characterised as a state of (locally) uniform field-line helicity.
CT11 Magnetohydrodynamics
Abdelhafid Younsi (Djelfa , Algeria)
Uniqueness result for Leray-Hopf weak solutions of the incompressible 3D Navier-Stokes equations
In this paper, we show the uniqueness of Leray-Hopf weak solutions to the 3-dimensional incompressible Navier-Stokes equations for any arbitrary initial data. First we prove the uniqueness in a bounded set B_{‖.‖}(u,δ(t)) of Leray-Hopf weak solutions. This set B_{‖.‖}(u,δ(t)) is a closed ball of radius δ(t) and centered at u. The radius δ(t) is a function of time. Then we extend this result to all solutions in the class of Leray-Hopf weak solutions.
Poster
Henggui Zhang (Manchester)
Development of 3D model of the heart for the study of atrial fibrillation
Atrial fibrillation (AF) is one of the most common cardiac arrhythmias causing morbidity and even mortality. However, possible mechanisms underlying the initiation and control of AF unclear yet. In this talk, I'll review the work on the development of 3D human heart, and its application to underpin molecular and ionic mechanisms for AF initiation and drug intervention.
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Yalin Zheng (Liverpool)
Embedding geometrical constraints in deep learning models for the segmentation of cardiac MRI images
MS 17 TBA
MS17 Progress and Trends in Mathematical Modelling of Cardiac Function
Kostas Zygalakis (Edinburgh)
Bayesian inverse problems, prior modelling and algorithms for posterior sampling
Bayesian inverse problems provide a coherent mathematical and algorithmic framework that enables researchers to combine mathematical models with data. The ability to solve such inverse problems depends crucially on the efficient calculation of quantities relating to the posterior distribution, which itself requires the solution of high dimensional optimization and sampling problems. In this talk, we will study different algorithms for efficient sampling from the posterior distribution under two different prior modelling paradigms. In the first one, we use specific non-smooth functions, such as for example the total variation norm, to model the prior. The main computational challenge in this case is the non-smoothness of the prior which leads to “stiffness” for the corresponding stochastic differential equations that need to be discretised to perform sampling. We address this issue by using tailored stochastic numerical integrators, known as stochastic orthogonal Runge-Kutta Chebyshev (S-ROCK) methods, and show that the corresponding algorithms are able to outperform the current state of the art methods. In the second modelling paradigm, the prior knowledge available is given in the form of training examples and we use machine learning techniques to learn an analytic representation for the prior. We exhibit numerically that this “data-driven” approach improves the perfomance in a number of different imaging tasks, such as image denoising and image deblurring.
MS02 Mathematics for Data Science
Maxim Zyskin (Oxford)
The Boltzmann equation in relation to Onsager–Stefan–Maxwell diffusion for Lennard-Jones gas mixtures
Charles W. Monroe. M. Zyskin
The Enskog method is an asymptotic method that provides solutions of the Boltzmann equation close to local equilibrium. The method predicts kinetic properties of gases, such as Stefan-Maxwell diffusivities, in terms of certain collision integrals. We investigate how those analytical predictions compare with molecular dynamics simulations for mixtures of Lennard-Jones gases and experiments with mixtures of monatomic gases. We apply the Enskog method to derive continuum-level model equations. Within the limitations of Enskog's method, these results identify with a set of multi-species transport equations that Goyal and Monroe recently derived using the alternative theory of irreversible thermodynamics.
CT 15 Statistical and Numerical Methods