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)

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.


Friday 21th February 2020, 2:00pm
Dr. Cesare Tronci, University of Surrey
Modeling efforts in hybrid kinetic-MHD theories of magnetized plasmas

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

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

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

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

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

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

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

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

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

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

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.

Friday 22nd November, 2:00pm
Prof. Adriana Dawes, Ohio State University
Antagonistic motor protein dynamics in contractile ring structures

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

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

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

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 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

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 pro les

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 infi nite 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 = εα 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 pro le 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

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

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

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

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?

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.