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

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

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

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

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

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

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 3rd May, 2:00pm
Prof. Angela Stevens, Universität Münster

Friday 10th May, 2:00pm
Dr. Elaine Crooks, Swansea University

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

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

Friday 31st May, 2:00pm
Dr. Helen Burgess, University of St Andrews

Friday 14th June, 2:00pm
Prof. David Rand, University of Warwick

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?

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

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

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

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

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

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.