# ABSTRACTS - MinisymposiUM

Minisymposium abstracts are available to download here

Abstracts arranged alphabetically by speaker surname.

Details of the minisymposium, venue and timing are provided after each abstract.

**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, Clyde 4, Friday*

**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, Hillhead 4, Friday*

**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, Bute 4, Tuesday*

**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, Bute 5, Thursday*

**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, Clyde 4, Friday*

**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, Clyde 5, Wednesday*

**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, Clyde 4, Tuesday*

**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, Clyde 4, Tuesday*

**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, Clyde 4, Thursday*

**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, Hillhead 5, Friday*

**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, Hillhead 5, Thursday*

**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, Bute 4, Friday*

**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, Bute 4, Tuesday*

**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, Clyde 4, Wednesday*

**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, Bute 5, Tuesday*

**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, Bute 5, Wednesday*

**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, Bute 5, Friday*

**Helen Burgess **(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, Hillhead 3, Wednesday*

**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, Kelvin 4, Tuesday*

**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, Hillhead 3, Thursday*

**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, Hillhead 5, Thursday*

**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, Hillhead 3, Friday*

**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, Clyde 5, Friday*

**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, Clyde 5, Tuesday*

**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, Clyde 4, Wednesday*

**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, Hillhead 3, Thursday*

**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, Clyde 4, Tuesday*

**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, Kelvin 5, Tuesday*

**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, Hillhead 4, Thursday*

**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, Bute 4, Friday*

**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, Hillhead 3, Wednesday*

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Kelvin 4, Tuesday*

**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, Clyde 4, Friday*

**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, Hillhead 4, Thursday*

**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, Bute 4, Tuesday*

**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, Hillhead 5, Wednesday*

**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, Bute 5, Wednesday*

**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, Bute 5, Friday*

**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, Clyde 5, Friday*

**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, Hillhead 3, Wednesday*

**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, Hillhead 5, Friday*

**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, Hillhead 4, Thursday*

**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, Kelvin 4, Tuesday*

**Sergei Fedotov **(Manchester)

*Front propagation for proliferating Lévy walkers*

Non-linear integro-differential kinetic equations for proliferating Lévy walkers have been developed. A hyperbolic scaling has been applied to these equations to get the Hamilton-Jacobi equations that allow to determine the rate of front propagation. We found the conditions for switching, birth and death rates under which the propagation velocity reaches the maximum value, i.e. the walker's speed.

*MS12 Front Propagation in PDE, probability and applications, Clyde 5, Wednesday*

**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, Hillhead 4, Thursday*

**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, Clyde 5, Thursday*

**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, Bute 5, Tuesday*

**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, Bute 5, Tuesday*

**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, Hillhead 3, Thursday*

**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, Clyde 4, Friday*

**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, Bute 5, Thursday*

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Bute 5, Wednesday*

**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, Clyde 4, Thursday*

**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, Hillhead 5, Wednesday*

**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, Kelvin 4, Tuesday*

**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, Bute 5, Tuesday*

**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, Clyde 5, Wednesday*

**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, Clyde 4, Wednesday*

**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, Hillhead 4, Friday*

**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, Bute 5, Tuesday*

**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, Clyde 4, Wednesday*

**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, Bute 5, Friday*

**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, Bute 5, Friday*

**David Henry **(Cork)

*Water wave profile recovery from pressure measurements at the seabed*

The reconstruction of the water wave surface profile from bottom pressure measurements is an important issue for marine engineering applications, but corresponds to a difficult mathematical problem. Measuring the surface of water waves directly is difficult and costly, particularly in the ocean, so a commonly employed alternative is to calculate the free-surface profile of water waves by way of the so-called pressure transfer function, which recovers the free-surface elevation using measurements from submerged pressure transducers. Until recent years, most studies were based on a linear transfer function, primarily due to the intractability of the nonlinear governing equations, and even then for irrotational waves. In this talk I outline some recent developments which expands the theory to incorporate weakly nonlinear rotational waves, and fully nonlinear irrotational waves (up to Stokes' extreme wave). This includes joint work with Gareth Thomas, and Didier Clamond.

*MS26 Recent Advances in Nonlinear Internal and Surface Waves, Hillhead 5, Thursday*

**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, Bute 5, Wednesday*

**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, Bute 4, Wednesday*

**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, Hillhead 5, Wednesday*

**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, Kelvin 5, Tuesday*

**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, Hillhead 3, Wednesday*

**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, Clyde 5, Thursday*

**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, Bute 5, Thursday*

**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, Bute 4, Thursday*

**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, Hillhead 5, Friday*

**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, Kelvin 5, Tuesday*

**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, Hillhead 4, Wednesday*

**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, Bute 4, Friday*

**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, Hillhead 4, Friday*

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Hillhead 5, Friday*

**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, Kelvin 5, Tuesday*

**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, Clyde 5, Wednesday*

**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, Clyde 4, Tuesday*

**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, Hillhead 4, Thursday*

**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, Bute 5, Friday*

**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, Clyde 5, Wednesday*

**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, Hillhead 3, Friday*

**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, Bute 5, Thursday*

**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, Hillhead 3, Thursday*

**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, Hillhead 5, Tuesday*

**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, Clyde 5, Friday*

**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, Bute 4, Wednesday*

**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, Clyde 4, Thursday*

**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, Kelvin 5, Tuesday*

**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, Bute 4, Tuesday*

**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, Hillhead 5, Wednesday*

**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, Bute 5, Friday*

**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, Clyde 4, Wednesday*

**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, Bute 4, Thursday*

**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, Bute 4, Thursday*

**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, Hillhead 3, Friday*

**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, Clyde 5, Wednesday*

**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, Bute 4, Thursday*

**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, Bute 5, Thursday*

**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, Clyde 5, Thursday*

**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, Clyde 5, Tuesday*

**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, Hillhead 5, Thursday*

**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.

**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, Kelvin 5, Tuesday*

**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, Hillhead 5, Friday*

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Hillhead 5, Tuesday*

**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, Hillhead 5, Tuesday*

**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, Hillhead 5, Tuesday*

**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.

**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, Clyde 5, Tuesday*

**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, Bute 4, Wednesday*

**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, Hillhead 4, Thursday*

**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, Clyde 4, Thursday*

**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, Clyde 5, Thursday*

**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, Bute 4, Tuesday*

**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, Hillhead 5, Tuesday*

**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, Bute 5, Wednesday*

**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, Clyde 4, Wednesday*

**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, Clyde 5, Wednesday*

**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, Clyde 5, Friday*

**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, Hillhead 3, Friday*

**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, Hillhead 3, Wednesday*

**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, Hillhead 5, Wednesday*

**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, Hillhead 5, Friday*

**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, Clyde 4, Thursday*

**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, Hillhead 5, Wednesday*

**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, Clyde 5, Tuesday*

**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, Bute 4, Wednesday*

**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, Clyde 5, Friday*

**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, Bute 4, Thursday*

** **

**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, Bute 4, Thursday*

**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, Hillhead 5, Thursday*

**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, Hillhead 3, Wednesday*

**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, Clyde 4, Friday*

**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, Hillhead 5, Thursday*

**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, Bute 5, Wednesday*

**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, Bute 5, Tuesday*

**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, Hillhead 3, Friday*

**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, Bute 4, Wednesday*

**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, Hillhead 4, Wednesday*

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Hillhead 4, Wednesday*

**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, Kelvin 4, Tuesday*

**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.

**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, Clyde 5, Tuesday*

**James Van Yperen **(Sussex)

*Modelling COVID-19 for Sussex healthcare demand and capacity*Abstract

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, Clyde 5, Thursday*

**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, Clyde 5, Tuesday*

**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, Bute 5, Thursday*

**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, Clyde 5, Thursday*

**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)

**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 moelling of liquid crystalline fluids, Hillhead 4, Tuesday*

**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, Hillhead 3, Friday*

**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, Clyde 5, Friday*

**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, Kelvin 4, Tuesday*

**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, Clyde 4, Friday*

**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, Hillhead 5, Tuesday*

**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).

**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, Bute 4, Tuesday*

**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.

**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, Clyde 4, Thursday*

**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, Hillhead 3, Thursday*

**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, Hillhead 3, Thursday*

**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, Bute 4, Wednesday*