# Abstracts -

# Poster Session

The Poster Session will run throughout the event. Each poster has a dedicated space on Sococo, where you can view the poster and speak with the poster presenters.

**All poster presenters are requested to be available in their rooms between 12.30 and 13.00 (BST) on Days 1 and 2. **

In addition, we have asked the poster presenters to maximise their availability over the lunchtime sessions on Days 1, 2 &3, and during the Happy Hour on Days 1 and 2. Remember you can use the 'find' function in Sococo to contact a poster presenter and arrange a time to discuss their poster.

Details of other activities scheduled over the lunchtime break each day are available here

## Abstracts

Contributed Poster abstracts are arranged alphabetically by surname.

Posters abstracts are available to download here

At the end of the abstract the Sococo floor the poster is located on is provided.

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**Thoraya**** Al****harthi**** **(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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**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.

*Venue: Glasgow Floor*

**Mohammed ****A****lsubhi **(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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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, 13^{th} 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).

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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)

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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)

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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.

*Venue: Bute Floor*

**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 Burger-Mozes groups, Leary-Minasyan groups and Sarithmetic lattices. Finally, we will present a number of applications.

*Venue: Bute Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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..

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**Gordon McNicol **(Glasgow)

*Self-excited oscillations in flow through a flexible-walled channel with a heavy wall*

We develop a model for laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a membrane of finite mass that is held under longitudinal tension. Driving the flow using an imposed upstream pressure, we employ a linear stability analysis to investigate the static and oscillatory global instabilities of the system, predicting the critical flow conditions required for the onset of self-excited oscillations. We show that the primary global instability of the system involves an oscillating wall profile with a single extremum. We further show that increasing the wall mass and membrane tension have contrasting influences on the oscillation frequency of this mode. We examine the asymptotic behaviour of the primary global mode in the limit where the wall mass and tension become large simultaneously and the oscillation frequency remains finite. In this regime we develop analytical expressions for the critical Reynolds number and oscillation frequency, which show excellent agreement with numerical results. Furthermore, we use a weakly non-linear analysis to elucidate the mechanism driving the instability: oscillation reduces the mean flow rate along the channel, reducing the overall dissipation and requiring less work done by upstream pressure than in the basic state.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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).

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Clyde Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**Chonilo Saldon **(Zamboanga del Norte Philippines)

*Diversity Indices and Species Modeling of Household Garden Plants*

Over the months of the imposition of community quarantine due to the COVID-19 pandemic, there has been a significant rise of plant collectors, growers and enthusiasts in the Philippines. While efforts of plants enthusiasts in propagation are relevant in the promotion of green living and eco-friendly community, it has also created challenges. Gathering and trading of wild plants from the forest including threatened species have been reported. This study categorizes household plants into critically endangered, endangered, vulnerable and other threatened species amongst others, computes the diversity indices and provides subsequent mathematical model. In particular, the study made use of Shannon-Weiner’s Species Diversity index, Pielou’s Index of Species Evenness and Margalef Index for Species Richness in recording diversity indices. Data were taken from 100 - 2mx2m household garden partitions. Implication of the study discussed are discussed in terms responsible plant propagation and species conservation.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**Elias Siguenza **(Birmingham)

*Feeding the habit: uncovering the metabolic relationship between bone marrow mesenchymal stems cells and malignant plasma cells in multiple myeloma.*

Multiple myeloma (MM) is an incurable malignant disease of plasma cells with the poorest 5-year survival of any haematological malignancy. Bone marrow (BM) residency of malignant plasma cells is an absolute requirement for their survival and proliferation, suggesting that the microenvironment within this niche is a critical driver of disease. We previously showed that the metabolism of the BM is significantly altered in patients with multiple myeloma, and that the BM mesenchymal stem cell (BMMSC), the major supportive cell type for malignant plasma cells, was significantly and irreversibly transformed. We hypothesise that these two cell types form a co-operative metabolic network within the BM that is critical for the survival and proliferation of malignant plasma cells. If true, then targeting this metabolic communication will directly impact on disease progression and response to therapy, improving patient outcomes. We created a testable model of the metabolic network formed by malignant plasma cells and BMMSCs. Using this we aim to identify enzymes or transporters that represent hubs, the inhibition of which would result in a breakdown of the community and sensitisation to standard therapeutic approaches to treating this incurable cancer.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*

**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.

*Venue: Hillhead Floor*