Welcome to the webpage of the Applied Mathematics Seminars of the University of St Andrews!

Unless otherwise indicated, the seminars will be held in Lecture Theatre D, Mathematical Institute (MI).

All interested are welcome to attend. Tea will be available afterwards in the Staff Common Room.

External visitors are advised to confirm arrangements with the Secretary: Tel. (01334) 463744/463747.

### Candlemas Semester Programme 2020

#### Upcoming lecture

In view of the current Coronavirus development and the policy of the University of St. Andrews, the upcoming lectures of the Applied Mathematics Seminar are postponed and will be held in a future date.May 1st: Prof. Ray Goldstein, University of Cambridge

May 15th: Prof. Alain Goriely, University of Oxford

May 22nd: Prof. Christian Beck, Queen Mary, University of London

#### Previous lectures

**Friday, March 13th 2020, 2:00pm**

Dr. Nikola Popovic, University of Edinburgh

**A Geometric Analysis of a Model for Micro-Electro Mechanical Systems (MEMS)**

Abstract:

*
Micro-Electro Mechanical Systems (MEMS) are deﬁned as very small structures that combine electrical and mechanical components on a common substrate; they have found numerous applications, including in medicine, optics, and telecommunications, and are encountered in a wide variety of technological devices nowadays.
Here, the electrostatic-elastic case is considered, whereby an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential. That case is commonly described by a parabolic partial differential equation that contains a singular nonlinear source term which can give rise to the “touchdown” phenomenon. Mathematically speaking, touchdown may imply the non-existence of steady states in the model, or blow-up of solutions in finite time. In a recently proposed, regularised model, a small “regularisation” parameter ε is introduced, whereby such singularities can be avoided by the introduction of an additional insulating layer between the membrane and the ground plate. Standard techniques from dynamical systems and geometric singular perturbation theory, in combination with the desingularisation technique known as “blow-up”, allow for a precise description of steady-state dynamics in the regularised model, as well as for a perturbative resolution of the resulting bifurcation diagram. The interplay between various model parameters is emphasised; in particular, the focus is on the singular limit as some of these parameters tend to zero.
*

#### Previous

**Friday 21th February 2020, 2:00pm**

Dr. Cesare Tronci, University of Surrey

**Modeling efforts in hybrid kinetic-MHD theories of magnetized plasmas**

Abstract:

*
Over the decades, multiscale modeling efforts have resorted to powerful methods, such as asymptotic/perturbative expansions and/or averaging techniques. As a result of these procedures, finer scale terms are typically discarded in the fundamental equations of motion. Although this process has led to well consolidated plasma models, consistency issues may emerge in certain cases especially concerning the energy balance. This may lead to the presence of spurious instabilities that are produced by nonphysical energy sources. The talk proposes alternative techniques based on classical mechanics and its underlying geometric principles. Inspired by Littlejohn's guiding-center theory, the main idea is to apply physical approximations to the action principle (or the Hamiltonian structure) underlying the fundamental system, rather than operating directly on its equations of motion. Here, I will show how this method provides new energy-conserving variants of hybrid kinetic-MHD models, which suppress the spurious instabilities emerging in previous non-conservative schemes.
*

**Friday 7th February 2020, 2:00pm**

Prof. Colin Torney, University of Glasgow

**From machine learning to migration: Understanding collective animal movement in ecology**

Abstract:

*
Recent advances in technology and quantitative methods have led to a growth in our ability to study mobile animal groups in their natural environments. Understanding the movement patterns of these groups requires the study of individual movement behaviour and the interactions between leadership, imitation, and environmental drivers that influence movement decisions. In this talk I will also discuss the methods we're using to investigate these questions in the field, including tools to collect video footage, computational methods to locate animals within images, and techniques to infer behavioral rules from movement data for both individuals (GPS collar data) and social groups (from UAV footage).
*

**Friday 31st January 2020, 2:00pm**

Prof. Nick Monk, University of Sheffield

**Modelling the effects of oscillatory transients in models of cell fate decisions**

Abstract:

*
In a wide range of biological systems, “decision making” can be understood as the process by which a multistable system approaches one of its stable equilibria from a given initial condition. When the system in question can naturally be decomposed into multiple units (e.g. cells in a multicellular tissue), decision making typically depends on interactions between these units (rather than being autonomous to each unit). I will present a model for bistable decision making based on a common signalling mechanism that operates between cells, and demonstrate that under biologically reasonable conditions, the model exhibits significant oscillatory transients. These transients introduce an additional time-scale into the dynamics of the model, which interacts with other relevant time-scales in the model to open up new population-level behaviours.
*

### Martinmas Semester Programme 2019

**Friday 27th September, 2:00pm**

Dr. Helen Burgess, University of St Andrews

**Vortex unclustering and statistical cooling in the turbulent inverse energy cascade**

Abstract:

*
The statistics of clustered and free vortices - rapidly rotating, long-lived patches of fluid -
in the inverse energy cascade of two-dimensional turbulence are studied.
Vortices are defined as 'clustered' if they rotate in the same direction and are contained within a contiguous
region bounded by a level set of the streamfunction, and if this region in addition contains no vortices
of the opposite sign. Vortices that do not meet these criteria are defined as 'free'.
The distribution of vortices across scales differs strikingly between the clustered and free populations.
The range in which n(A) ∼ A ^{ −1 }, present at intermediate scales in the full number density (Burgess & Scott 2017),
is found only within the clustered population, while the free number density scales as n(A) ∼ A^{ −3 }.
The scaling ranges in the number density found by Burgess & Scott (2017) are thus associated with vortices located in different flow regions.
The number of clustered vortices falls off in time, while the number of free vortices increases. At the same time, the average size of
clustered vortices grows in time, while the average size of free vortices decreases, showing that smaller vortices are preferentially
ejected from clusters. This is consistent with the system meeting energy constraints while maximising randomness, and suggests a trend
toward unclustering or 'statistical cooling' of the vortex population.
*

**Friday 4th October, 2:00pm**

Dr. Ruth Bowness, University of St Andrews

**Exploring post-primary infection in Mycobacterium tuberculosis using a hybrid discrete-continuum cellular automaton model**

Abstract:

*
Tuberculosis (TB) is an infectious bacterial disease caused by Mycobacterium tuberculosis.
Despite significant recent advances, TB is the biggest infectious killer globally with someone dying from the disease
every 18 seconds. When Mycobacterium tuberculosis bacteria enter the lungs, a complex immune response ensues and results
in the formation of granuloma structures. When these granulomas are unable to contain the bacteria, active disease develops.
At different degrees of disease severity, patients seek medical assistance, after which antibiotics are prescribed.
The degree of antibiotic penetration into and through the granuloma is uncertain. The outcome of treatment is complicated
by dormancy when the bacteria become temporarily resistant to antibiotics. We have developed a hybrid discrete-continuum
cellular automaton model to study disease progression and treatment in the lung. The model contains discrete agents, or individuals,
which model the spatio-temporal interactions (migration, binding, killing etc.) of bacteria, macrophages and T cells.
The spatial movement of cells is governed by biased random walks, while the various cell-cell and cell-bacteria interactions
are governed by cellular automaton rules. Chemokine diffusion, oxygen diffusion and a Pharmacokinetic/Pharmacodynamic model is
also incorporated in the model via the numerical solution of appropriate PDEs. Several definitions and theories regarding
bacterial dormancy exist in the literature. In this work, we use our hybrid cellular automaton model to explore several concepts
of dormancy and their effect on treatment outcome.
*

**Friday 11th October, 2:00pm**

Dr. David Rees Jones, University of St Andrews

**Solidification and convection in mushy layers: implications for sea ice**

Abstract:

*
Seawater, metal alloys, and rocks are multi-component systems. When such systems solidify, they form a mushy layer,
which is a reactive multi-phase material consisting of a solid matrix with liquid in the interstices of the matrix.
Liquid can flow through the system, and a buoyancy gradient can drive convective flow. I discuss the linear and nonlinear
development of this convective flow. The fully developed state is particularly interesting, because the porous matrix can
be dissolved such that flow is concentrated in purely liquid channels. I discuss some simple models of the fully developed
state in both two and three dimensions and determine the rate of chemical transport in terms of a mush Rayleigh number and
the other dimensionless parameters of the system. Finally, I apply these mushy-layer models to propose improvements to the
representation of sea ice in climate models.
*

**Friday 18th October, 2:00pm**

Dr. Ben Goddard, University of Edinburgh

**Dynamic Density Functional Theory: Modelling, Numerics and Analysis**

Abstract:

*
In recent years, a number of dynamic density functional theories (DDFTs) have been developed, originally to describe
colloidal particles, but also with applications to cell dynamics and other areas of mathematical biology.
These DDFTs aim to overcome the high-dimensionality of systems with large numbers of particles by reducing to the
dynamics of the one-body density, described by a non-local, nonlinear PDE in only three spatial dimensions, independent
of the number of particles.
The standard derivations start from stochastic equations of motion, but there are fundamental differences in the
underlying assumptions in each DDFT. I will begin by giving an overview of some DDFTs, highlighting the assumptions
and range of applicability. Particular attention will be given to the inclusion of inertia and hydrodynamic interactions,
both of which strongly influence non-equilibrium properties of the system. I will then demonstrate the very good agreement
with the underlying stochastic dynamics for a wide range of systems, including confined systems with hydrodynamic interactions.
If time allows, I will also discuss an accurate and efficient pseudospectral numerical code that we have developed, as well as
the passage to Smoluchowski-like and Navier-Stokes-like equations in appropriate limits.
Joint work with Serafim Kalliadasis, Rory Mills-Williams, Greg Pavliotis, and Andreas Nold.
*

**Friday 25th October, 2:00pm**

Dr. Anirban Guha, University of Dundee

**Predicting vortex merging and ensuing turbulence characteristics in shear layers from initial conditions**

Abstract:

*
Unstable shear layers in environmental and industrial flows roll up into a series of vortices,
which often form complex nonlinear merging patterns like pairs and triplets.
These patterns crucially determine the subsequent turbulence, mixing and scalar transport.
We show that the late-time, highly nonlinear merging patterns are predictable from the linearized initial state.
The initial asymmetry between consecutive wavelengths of the vertical velocity field provides an effective measure of
the strength and pattern of vortex merging. The predictions of this measure are substantiated using direct
numerical simulations. We also show that this measure has significant implications in determining the route to
turbulence and the ensuing turbulence characteristics.
*

**Friday 1st November, 2:00pm**

Dr. Heiko Gimperlein, Heriot-Watt University

**Nonlocal diffusion in biological and robotic systems**

Abstract:

*
This talk discusses diffusion processes beyond Brownian motion and their description by nonlocal differential operators,
such as the fractional Laplacian. The long range movement of certain organisms in the presence of a chemoattractant can be
governed by long distance runs, according to an approximate Levy distribution. Starting from a microscopic velocity-jump model
for the movement, we derive nonlocal Patlak-Keller-Segel equations for the macroscopic evolution of the density. Their analysis
allows to develop efficient numerical methods for their simulation. We consider applications to the chemotactic movement of E. coli
and T-cells, as well as to the design of swarm robotic systems.
*

**Friday 8th November, 2:00pm**

Prof. Rachel Norman, University of Stirling

**Using mathematical models to understand tick borne pathogen dynamics and control in a multi-host system with multiple transmission routes**

Abstract:

*
Ticks have complex lifecycles which mean that they feed on multiple host individuals and host species.
They can transmit several different diseases including Lyme disease and Tick borne encephalitis. In this talk I will focus on Louping Ill which is a tick borne infection of sheep and grouse.
I will present a series of mathematical models of the transmission and dynamics of this pathogen which have been developed over a 20 year time period to answer a range of different biological questions.
*

**Friday 15th November, 2:00pm**

Dr. Lyuba Chumakova, University of Edinburgh

**Why are we not falling apart: cytoskeleton self-organization and some results on intracellular transport**

Abstract:

*
For the correct cellular and therefore organism function, cellular components must be robustly
delivered to their biologically relevant location. This is achieved through intracellular transport,
where vesicles and organelles are transported like cargo via cars (molecular motors) along highways
(the microtubule cytoskeleton). Failure of this process can result in pathologies. I will present a study of intracellular
transport in epithelium, one of the four fundamental tissue types in all animals.
To understand the outcome intracellular transport, several fundamental questions stand out:
*

- Does the microtubule road-network have any order, do the microtubules self-organise?
- While organisms live in rapidly varying environments, their development and tissue properties are robust. Is the microtubule self-organisation robust as well?
- How to determine the motor-type (Kinesin or Dynein) that delivers a particular cargo by only knowing the final cargo location in the cell?
- Why are we not falling apart? The transmembrane protein responsible for cell-cell adhesion is E-cadherin, and the adhesion strength is determined by the amount of E-cadherin on the cell boundary. How is intracellular transport of E-cadherin re-organised during the Drosophila embryo development?

I will present the mathematical models and stochastic simulations which suggested the answers to these questions, how this led to setups of novel biological experimental which verified our hypotheses, and finally develop a sequence of simple models uncovering the mathematical basis of the underlying biological phenomena.

I will present the mathematical models and stochastic simulations which suggested the answers to these questions, how this led to setups of novel biological experimental which verified our hypotheses, and finally develop a sequence of simple models uncovering the mathematical basis of the underlying biological phenomena.

**Friday 22nd November, 2:00pm**

Prof. Adriana Dawes, Ohio State University

**Antagonistic motor protein dynamics in contractile ring structures**

Abstract:

*
Stable ring-shaped contractile structures play important roles in biological processes including material transport. Many of these contractile structures rely on motor proteins called myosins for establishing and maintaining their geometry. We investigate force generation by the Type II myosins NMY-1 and NMY-2 using ring channels in the nematode worm C. elegans as our model system.
By exploiting the ring channel's circular geometry, we derive a second order ODE to describe the evolution of the radius of the ring channel. By comparing our model predictions to experimental depletion of NMY-1 and NMY-2, we show that these myosins act antagonistically to each other, with NMY-1 exerting force orthogonally and NMY-2 exerting force tangentially to the ring channel opening.
I will discuss early efforts to characterize the kinetic properties of NMY-1 and NMY-2 using individual based modelling of motor protein activity, and the emergence of ring-like structures using topological data analysis.
*

**Friday 29th November, 2:00pm**

Dr. Jochen Kursawe, University of St Andrews

**Quantitative modelling of embryonic development**

Abstract:

*
The study of morphogenesis promises to shed light on developmental diseases and to pave the way for the growth of artificial organs.
Recent years have seen a rise in quantitative data for many embryonic processes. However, these new data lead to challenges at each stage
of the scientific method, including the design of quantitative hypotheses as well as data analysis and data interpretation.
Here, I will present projects that illustrate how mathematical methods can help overcome challenges in the quantitative study of embryonic development.
I will show how multi-scale, cell-based models can be designed to make experimentally testable predictions on tissue growth.
I will present a novel algorithm that uses graph theoretic concepts to enable cell-tracking in live-imaging microscopy videos.
Finally, I will discuss how Bayesian inference can provide insights into the mechanics of tissue growth and into the roles of gene expression
dynamics during cell differentiation.
*

**Friday 13th December, 2:00pm**

Dr. Ioannis Markou, Foundation for Research and Technology Greece, Crete

**Averaging Models in the Study of Collective Behavior and Emergence**

Abstract:

*
Collective crowd dynamics has attracted major attention in the past decades. Several models have been introduced to describe emergent behavior in systems of rational agents, groups of
animals, and automatons. Here we present a class of models that follow an underlying averaging principle to reach consensus or otherwise more complex long time behavior.
Such models include opinion models, like the Hegselmann-Krause model. Models for groups of animals that flock, swarm or mill e.g. the Vicsek, Inertial-Spin, and Cucker-Smale models. Models for automated vehicles e.g. control problems. Synchronization in biological systems (the Kuramoto model).
Learning and memory storage (artificial neural networks), and many others. A particularly successful model is the Cucker-Smale model of flocking which was introduced in 2007 and predicts alignment of velocities in animal crowds. Despite its phenomenal simplicity several variations of
the C-S model have since been considered to make this model more realistic. In this talk, we review some of the most important recent developments in collective dynamics with extra weight put on the
Cucker-Smale alignment model and contributions by the presenter. We also discuss some open problems.
*

### Candlemas Semester Programme 2019

**Friday 15th February, 2:00pm**

Dr. Philip Murray, University of Dundee

**Genetic oscillations in the developing embryo: theory meets experiment**

Abstract:

*
Somitogenesis is a process that occurs during the development of the vertebrate embryo.
At regular intervals in space and time, a pair of segments is formed, one on either side of
the future spinal chord. Underlying the oscillatory pattern formation is a multicellular,
genetic oscillator known as the somitogenesis clock that has a period of the order of hours.
Mathematical models of spatiotemporal dynamics of gene expression, often formulated using
partial differential equations, have a played a crucial role in understanding key features of pattern formation.
In this talk I will describe experiments undertaken at the University of Dundee in which
the spatio-temporal dynamics of gene expression are measured in a piece of embryonic tissue.
After showing that the experimental data are noisy and non-stationary, we will explore the following questions:
(i) how do we extract variables from the experimental data that allow mathematical models to be validated?;
and (ii) can current models explain the spatiotemporal dynamics observed in the experiments?
*

**Friday 22nd February, 2:00pm**

Dr. Nikolaos Sfakianakis, University of St Andrews

**The role of Numerical Analysis in Mathematical Biology: some examples, difficulties, and cavities**

Abstract:

*
The reproducibility of results in Mathematical Biology has recently
become a very active topic of discussion and criticism. In this
discussion, the role of Numerical Analysis is of principal importance.
It is the first tool used to analyse the mathematical models and verify
their biological validity, and to extract, quantify, and communicate the
results.
In this talk, I present several examples from my work where elaborate
numerical methods and techniques where able to shed light on questions
evading the usual numerical treatment. I address the difficulties in
devising, implementing, and analysing these methods, and discuss some of
their cavities.
*

**Friday 1st March, 2:00pm**

Dr. Mauricio González-Forero, University of St Andrews

**Inferring how the human brain evolved**

Abstract:

*
A longstanding fascination has been to identify why the human brain evolved.
Despite much interest, it has been difficult to address this question with previously available research tools.
To circumvent this problem, we have developed mathematical models that enable one to study in silico how the human brain
could have evolved. The models are based on evolutionary invasion analyses for function valued traits in an age structured
population, and are formulated using metabolic theory which enables parameter estimation from available empirical data.
Application of the models so far supports hypotheses postulating ecological problem solving (e.g., for foraging or food processing)
rather than social problem solving (e.g., for cooperating or competing with peers) as a key driver of human brain expansion,
in contrast to commonly held views. In this talk, I will provide an overview of the models and their results.
*

**Friday 8th March, 2:00pm**

Dr. Christina Cobbold, University of Glasgow

**Modelling the phenological effects of environmental drivers on mosquito abundance: implications for West Nile virus transmission**

Abstract:

*
Mosquito-borne diseases cause substantial mortality and morbidity
worldwide. These impacts are widely predicted to increase as temperatures
warm, since mosquito biology and disease ecology are strongly linked to
environmental conditions. However, direct evidence linking these changes
to mosquito-borne disease is rare, and the ecological mechanisms that may
underpin such changes are poorly understood. I focus on West Nile virus
(WNV), a mosquito-borne arbovirus infecting avian hosts, that can spill
over into humans. Outbreaks of WNV are common in Africa, and Southern and
Eastern Europe, with recent outbreaks reported in France and Spain. There
has yet to be an outbreak in the UK, but there is current concern that
passerine migratory bird species could introduce the disease northward.
However, the question remains, if WNV is introduced in the UK, can the
disease establish? I present a mechanistic environmentally-driven
stage-structured host-vector mathematical model for predicting the
seasonal dynamics of WNV in current and future climates in the UK. The
model predicts that WNV is unlikely to establish in the foreseeable
future, although climate change is likely to increase the risk, with only
extreme climate predictions leading to possible WNV outbreaks.
*

**Friday 15th March, 2:00pm**

Dr. Craig Johnston, University of St Andrews

**Modelling Coronal Loops in the Solar Atmosphere**

Abstract:

*
The brightness of the emission from coronal loops in the solar atmosphere is strongly dependent on the temperature and
density of the confined plasma. After a release of energy, these loops undergo a heating and upflow phase, followed by a
cooling and downflow cycle. Throughout, there are significant variations in the properties of the coronal plasma.
In particular, the increased coronal temperature leads to an excess downward heat flux into the transition region (TR).
The plasma is unable to radiate this excess conductive heating and so the gas pressure increases locally.
The resulting pressure gradient drives an upflow of dense material, creating an increase in the coronal density.
This density increase is often called chromospheric evaporation. A process which is highly sensitive to the TR
resolution in numerical simulations. If the resolution is not adequate, then the downward heat flux jumps over the
TR and deposits the heat in the chromosphere, where it is radiated away. The outcome is that with an under-resolved TR,
major errors occur in simulating the coronal density evolution and, thus, the predicted loop emission. I will present
a new method that addresses the difficulty of obtaining the correct interaction between the corona and corona/chromosphere
interface. In the transition region, an adaptive thermal conduction approach is used that broadens any unresolved parts
of the atmosphere. I will show that this approach, referred to as TRAC, successfully removes the influence of numerical
resolution on the coronal density response to heating while maintaining high levels of agreement with fully resolved models.
*

**Friday 5th April, 2:00pm**

Dr. Cicely Macnamara, University of St Andrews

**Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue**

Abstract:

*
The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. One of the main Hallmarks of Cancer (Hanahan & Weinberg, 2000; 2011) is tissue invasion and metastasis. Mathematical modelling and simulation can complement traditional biological and experimental approaches to cancer research. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment. We are developing a novel model which allows one to simulate the behaviour of and spatio-temporal interactions between cells, blood vessels and other components of the tumour microenvironment. We use a 3D individual-based force-based model, i.e. each element (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. In this way we are able to reproduce, in silico, complex features of tumour development such as growth around a blood-vessel network or along the striations of fibrous tissue. As well as the mechanical interactions we also consider chemical interactions. For example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. In this talk I will present the current state of the art of the model and its capabilities.
*

**Friday 12th April, 2:00pm**

Prof. Bernard Legras, École Normale Supérieure Paris

**Transport properties and impact of the convection in the tropical tropopause layer during Asian monsoon**

Abstract:

*
Asian monsoon is the prominent convective region during summer and is
also the most polluted region on the Earth with potential global impact
of the emission. We consider how the air pracel detrained at high
altitude by convective clouds over Asian monsoon area are transported
partly in the troposhere by the Hadley-Walker circulation and partly in
the stratosphere in the ascending branch of the Brewer-Dobson. We show
how the Asian monsoon anticyclone confines parcels for some time over
Asia. We discuss also some salient results of the recent StratoClim
airborne campaign from Katmandu. The results have been obtained with
massive Lagrangian calculations based on the ERA-Interim and ERA5 data
with both kinematic and diababtic trajectories and provide an assessment
of these datasets.
*

**Friday 26th April, 2:00pm**

Prof. Sílvia Cuadrado, Universitat Autònoma de Barcelona

**On selection mutation equations: singular solutions and asymptotic proles**

Abstract:

*
Selection mutation equations are mathematical models of Darwinian evolution.
The selection term comes from an underlying ecological model for
an unstructured population where the evolutionary trait plays the role of
a parameter affecting vital rates. Existence of nontrivial equilibria of these
equations is shown using infinite dimensional versions of the Perron Frobenius theorem.
These equilibria tend to concentrate, when the mutation rate is small, at the so-called (in the sense of Adaptive Dynamics)
evolutionarily stable strategy of the underlying ecological model.
We will apply these results to a model for the maturation age and to
a model for clonal evolution of leukemic stem cells.
On the other hand, we will study the behavior for large time and small
mutation rate of a selection-mutation-competition model for a population
structured with respect to a phenotypic trait. We will analyze the interplay between
the time variable t and the rate ε of mutations showing that,
depending on α > 0, the limit ε → 0 with t = ε ^{α} can lead to population number
densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).
So, on the one hand, we analyze transient dynamics, which could be important in many biological situations (invasions, infections...)
and on the other hand we determine the asymptotic prole of the densities which shows heavy tails.
This could also be relevant for the survival of the population under environmental changes.
*

**Friday 3rd May, 2:00pm**

Prof. Angela Stevens, Universität Münster

**Sorting phenomena in interacting cell systems**

Abstract:

*
Attraction and repulsion between interacting cells with slight
differences in behavior can result in sorting phenomena, as they are observed in
developmental processes in biology. In this context energy functionals which describe
diffusive motion, short-range repulsion and long-range attraction are analyzed w.r.t. the structure of their minimizers.
Joint work with M. Burger, M. Di Francesco, and S. Fagioli.
*

**Friday 10th May, 2:00pm**

Prof. Elaine Crooks, Swansea University

**Invasion speeds in a competition-diffusion model with mutation**

Abstract:

*
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. After first
showing that, under reasonable conditions on the mutation and competition parameters,
the spreading speed of the two phenotypes is indeed determined by the
linearisation 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).
*

**Friday 17th May, 2:00pm**

Prof. Jan-Bert Flór, Laboratoire des Écoulements Géophysiques et Industriels

**Focusing internal waves and wave breaking in stratified and/or rotating fluids**

Abstract:

*
After introducing different cases of wave focusing, we discuss waves emitted by the oscillation of a horizontal torus in a
stably stratified and/or rotating fluid. The thus generated wave field has the form of a cone below and above the torus,
with a focal point in the apex where the waves reach a maximum amplitude. After considering the basic properties of this
wave field and a comparison with an approximate linear theory, we consider its nonlinear aspects for different amplitudes
of oscillations. A new non-dimensional number that is based on heuristic arguments allows us to characterise the wave field
and wave breaking in the focal region. Above a certain threshold of this focusing number, wave breaking appears for a
Richardson number equal to 0.25, and coincides with resonant wave triads appearing near the focal region
(Ermanyuk et al, J Fluid Mech 2017; Shmakova & Flor J. Fluid Mech 2019). Next we discuss the flow induced by the
oscillation of a vertically oscillating torus in a rotating fluid, and the formation of an isolated blob of turbulence
in the focal point. Whereas the turbulent blob in the rotating fluid leads to the formation of an organised columnar
vortex (see Duran Matute et al 2013, Phys. Rev. E 87), the overturning waves in a stratified fluid hardly organise
into a large scale motion. We discuss the differences in dynamics when stratification and/or rotation are present.
*

**Friday 24th May, 2:00pm**

Dr. Florent Michel, Durham University

**Nonlinear zero-mode and analogue Hawking radiation**

Abstract:

*
In 1981, William Unruh uncovered a mathematical correspondence between sound waves in a transcritical flow and scalar fields near a black hole, motivating that puzzling aspects of field theories in a black-hole space-time should have an analogue in acoustics. This analogy has then been extended to many different physical systems, from water surface waves to quantum fluids, often collectively referred to as “analogue gravity”. The main aim of this field is to detect the analogue of Hawking radiation, the elusive emission of particles from the near-horizon region of a black hole from quantum fluctuations. In this talk, I will first give a general introduction to analogue gravity, underlining its motivations, interest, and limitations, and summarise the experimental state of the art. I will then focus on the interplay between analogue Hawking radiation and resonance with a zero mode in hydrodynamics, how it affects the analogy, and the bounds it places on experimental parameters.
*

**Friday 7th June, 2:00pm**

Dr. Álvaro Viúdez Lomba, Institut de Ciències del Mar

**Some stable modes in two-dimensional and three-dimensional (geophysical) vortices**

**Friday 14th June, 2:00pm**

Prof. David Rand, University of Warwick

**Clocks & Cancer: Thinking multidimensionally**

### Martinmas Semester Programme 2018

**Friday 12th October, 2:00pm**

Dr. Giorgos Minas, University of St Andrews

**How does the noisy NF-kB signalling pathway distinguish between simultaneously received signals?**

Abstract:

*Cells constantly receive a multitude of different signals from their external environment.
They use networks of interacting molecules to respond to these signals and trigger the appropriate
actions. An important target of molecular biology is to identify and study the key components of
these networks that are often found to be therapeutic targets. An important example is the NF-κB
protein complex that is found to respond to a variety of different signals related to stress and
inflammation in order to activate a large number (>500) of different genes including those regulating
the immune system. The NF-κB network is noisy and complex with oscillatory dynamics involving multiple
feedback loops and therefore it is mathematically very interesting. In this talk, I am going to
introduce the NF-kB signalling pathway, discuss stochastic models describing its dynamics and then
attempt to develop a mathematical framework for assessing its ability to distinguish between
simultaneously received signals.*

**Friday 19th October, 2:00pm**

Dr. Gergely Röst, University of Oxford

**Time delays in mathematical biology and a case study from cell biology**

Abstract:

*First we give an overview how various types of delays (discrete, multiple, distributed, infinite, state-dependent) arise in problems related to mathematical epidemiology, physiology and population dynamics, and discuss the challenges in the analysis of nonlinear functional differential equations. Then, motivated by the go or grow type behaviour of glioma cells, from an individual based stochastic model we derive, as a mean field approximation, a new delay logistic equation with both discrete and distributed delays. A complete description of the global attractor will be given, and we show that very long transients exist with oscillatory patterns of various shapes.
*

**Friday 2nd November, 2:00pm**

Dr. Evgeny Ryzhov, Imperial College London

**Data-driven stochastic emulation of multi-scale geophysical flows**

Abstract:

*The problem of emulating eddy-scale features abounding in the ocean
has drawn a lot of attention recently. One of the reasons is that the
climate modelling community understands now that the brute-force
approach to resolve the fastest scales associated with meso- and
sub-mesoscale processes is computationally infeasible when dealing
with climate-scale time series. Another one is of a more theoretical
perspective such as whether it is possible to separate reliably scales
of motion and then to emulate a portion of these scales (or all of
them) by some stochastic models. In this talk, a method to decompose
geophysical flows into temporal scales is presented and applied to
idealised flows. The method we use is the Data-Adaptive Harmonic
Decomposition (DAHD). This method decomposes the given data into
temporal scales embedded within a chosen time-window based on the all
cross-correlations between spatial and temporal data points. The data
we use comes from two idealised flows: a wind-forced double gyre flow
and zonal stream flow in a channel. After applying the DAHD to the
data, the scales of motion are clearly separated revealing the complex
temporal structure of the double-gyre flow (there is a pronounced
low-frequency variability) and a relatively simple temporal structure
of the zonal stream flow (no low-frequency variability). After
decomposing the flow, variants of the multilayer stochastic modelling
techniques (MLSM) are implemented to give a satisfactory
correspondence between the original and emulated flows even for the
case of the double-gyre flow with involved temporal structure.*

**Friday 16th November, 2:00pm**

Prof. Nicola Bellomo, Politecnico di Torino

**New Trends Towards a Mathematical Sociology - A New Frontier of Applied Mathematics**

Abstract:

*A radical philosophical change that is unfolding in social and economic disciplines
and it is rapidly developing offering to applied mathematicians a number
of challenging modeling, analytic, and computational problems. Roughly
speaking, the new emerging point of view is characterized by an interplay among
Economics, Psychology, and Sociology, which is no longer grounded on the traditional
assumption of rational socio-economic behavior. While the rationale
for that approach, namely that Economics can be highly affected by heterogeneous
individual (rational or irrational) behaviors, reactions, and interactions
is widely accepted.
This seminar aims at presenting a critical overview on the state of the art focusing
on conceivable contributions of mathematical sciences. The presentation
focuses on the development of the mathematical kinetic theory and theoretical
tools of stochastic game theory to modeling the dynamics in sociology and
economy referring to large systems of interacting living entities.
Mathematical approach and tools are applied to the study of the competition
between criminality and security services.
G. Ajmone Marsan, N. Bellomo N., and L. Gibelli, Stochastic evolutionary
differential games toward a systems theory of behavioral social dynamics, Math.
Models Methods Appl. Sci., 26, 1051-1093, (2016).
D. Burini, S. De Lillo, and L. Gibelli, Stochastic differential "nonlinear" games
modeling collective learning dynamics, Phys. Life Rev., 16, 123-139, (2016).
D. Burini, L. Gibelli, and N. Outada, "A kinetic theory approach to the modeling
of complex living systems", in Active Particles, Vol. 1, Birkäuser, New
York, 229-258, (2017).*

**Friday 30th November, 2:00pm**

Prof. Pierre Degond, Imperial College London

**Models of emergent networks**

Abstract:

*In this talk, we will present a modelling framework for the
emergence of networks and their evolution. We will provide various examples
of such emergent networks: ant trails, extracellular fibers and blood
capillaries. We believe this framework can apply to other types of networks
in which the topology and topography of nodes and links is fuzzy and
evolutive.*

**Friday 14th December, 2:00pm**

Prof. Taoufik Hmidi, Universite de Rennes I

**Non uniform relative equilibria for Euler equations**

Abstract:

*We shall deal in this talk with non uniform rotating vortices for
planar Euler equations. We propose to give a general approach to
construct some of them near radial solutions. We provide a complete
study for the truncated quadratic profile and explore the rarefaction
of the bifurcating curves with respect to the parameters of the
profile. This is a joint work with Claudia Garcia and Juan Soler.*