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 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 31st May, 2:00pm**

Dr. Helen Burgess, University of St Andrews

**TBA**

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

Dr. Álvaro Viúdez Lomba, Institut de Ciències del Mar

**TBA**

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

Prof. David Rand, University of Warwick

**TBA**

### 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.*