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2016/17 semester 1 events in Mathematics

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Mon 19/09/16 15:00
Fulton G20
Mathematics Seminar
Dr. Nikola Popovic (University of Edinburgh)
A geometric classification of travelling front propagation in the Nagumo equation with cut-off
abstract

Abstract

Reaction-diffusion systems are frequently employed in the continuum approximation of discrete (many-particle) models; however, the quality of the approximation deteriorates when the number of particles is not sufficiently large. The (stochastic) effects of that discreteness have been modelled via the introduction of (deterministic) "cut-offs" that deactivate the reaction terms whenever the particle concentration is below a certain threshold. We present an overview of the effects of such a cut-off on the front propagation dynamics in the classical Nagumo equation. Our analysis is based on the method of geometric desingularisation ("blow-up"), in combination with dynamical systems techniques such as invariant manifolds and normal forms. In particular, our approach allows us to determine rigorously the asymptotics of the correction to the front propagation speed that is due to a cut-off. Moreover, it explains the structure of that asymptotics (logarithmic, superlinear, or sublinear) in dependence of the front propagation regime. Finally, it enables us to calculate the corresponding leading-order coefficients in the resulting expansions in closed form.

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Mon 26/09/16 15:00
Fulton G20
Mathematics Seminar
Dr. Agis Athanassoulis (University of Leicester)
On the XFEL Schrodinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
abstract

Abstract

The European X-ray Free Electron Lasers (XFEL), currently under construction, is designed to create pulses several orders of magnitude brighter, but also shorter in duration, than existing lasers. Using the X-ray flashes of the European XFEL, scientists will be able to map the atomic details of viruses, decipher the molecular composition of cells, take three-dimensional images of the nanoworld, film chemical reactions, and study processes such as those occurring deep inside planets. A side of this activity is the development of efficient and reliable models that will help design the experiments. This involves the study of a singularly perturbed nonlinear Schrodinger equation with fast timescales, the approximation of which by an effective slow-time model is desirable. The main mathematical ideas behind the construction of the effective model are outlined, along with some complementary numerical results. Finally some open questions are discussed, related to how the fine structure of the undulator (the device responsible for the singular perturbation) may affect the features of the laser beam.

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Mon 03/10/16 15:00
Fulton G20
Mathematics Seminar
Mr. Craig Johnston (University of St Andrews)
A New Approach for Modelling Chromospheric Evaporation in Response to Enhanced Coronal Heating.
abstract

Abstract

We present the results of 1D field-aligned simulations of the coronal plasma response to impulsive heating events. During these events, an increase in the coronal density occurs because the increased coronal temperature leads to an excess downward heat flux that the transition region (TR) is unable to radiate. This creates an enthalpy flux from the TR to the corona. The density increase is often called chromospheric evaporation. Sufficiently high resolution of the TR is essential in numerical simulations in order to obtain the correct coronal density (Bradshw & Cargill, ApJ, 2013). 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. Bradshaw & Cargill showed that major errors in simulating the coronal density evolution will occur. Therefore, to compensate for the jumping of the heat flux, when coarse resolutions are used, we propose that the TR should be treated as a discontinuity. We show that, by modelling the TR with an appropriate jump condition, we can remove the influence of poor numerical resolution and obtain the correct coronal density even when using resolutions that are compatible with 3D MHD simulations.

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Mon 10/10/16 15:00
Fulton G20
Mathematics Seminar
Dr. Filippo Cagnetti (University of Sussex )
The rigidity problem for symmetrization inequalities
abstract

Abstract

We will discuss several symmetrizations (Steiner, Ehrhard, and spherical symmetrization) that are known not to increase the perimeter. We will show how it is possible to characterize those sets whose perimeter remains unchanged under symmetrization. We will also characterize rigidity of equality cases. By rigidity, we mean the situation when those sets whose perimeter remains unchanged under symmetrization, are trivially obtained through a rigid motion of the (Steiner, Ehrhard or spherical) symmetral. We will achieve this through the introduction of a suitable measure-theoretic notion of connectedness, and through a fine analysis of the barycenter function for a special class of sets. These results are obtained together with several collaborators (Maria Colombo, Guido De Philippis, Francesco Maggi, Matteo Perugini, Dominik Stoger)

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Mon 17/10/16 15:00
Fulton G20
Mathematics Seminar
Dr. Ostap Hryniv (Durham University)
Re-entrant transition in a model of microtubule growth
abstract

Abstract

We introduce a Markov process in the space of strings made of two types of particle, whose dynamics imitates that of microtubules. The long term behaviour of this one-dimensional growth model is described in terms of the velocity of the "active end", which is an analytic function of the jump rates. The model exhibits a phase transition in that in some part of the parameter space the velocity possesses natural monotonicity properties, while in another region that monotonicity does not hold; in particular, increasing the local shrinking rate leads to a faster global growth.

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Mon 24/10/16 15:00
Fulton G20
Mathematics Seminar
Prof. Thomas Wihler (University of Bern)
Fully Adaptive Newton-Galerkin Methods for Semilinear Problems
abstract

Abstract

In this talk we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. An glimpse on time-dependent problems will be as well.

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Mon 31/10/16 15:00
Fulton G20
Mathematics Seminar
Dr. Chandrasekhar Venkataraman (University of St. Andrews)
Modelling and simulation of phenotypic heterogeneity and drug resistance in solid tumours
abstract

Abstract

A growing body of evidence suggests that the emergence of phenotypic heterogeneity and drug resistance in tumours is due to a process of adaption or evolution. Specifically, it is hypothesised that spatial variations in the concentration of abiotic factors within the tumour environment may create local niches driving natural selection. In this talk we investigate the validity of such a hypothesis by developing a mathematical model linking the dynamics of a structured tumour cell population with the concentrations of abiotic factors in the microenvironment. For biological relevance we use parameters in the model that are obtained from experimental data and we pose the system on real 3D tumour geometries obtained from image data. We compare numerical simulations of the full model using a finite element approximation with semi-analytical results on the asymptotically selected trait and observe good agreement between the two.

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Mon 14/11/16 15:00
Fulton G20
Mathematics Seminar
Prof. Thomas Richter (University of Magdeburg)
Computational Solid Mechanics and Fluid-Structure Interactions for Biological and Medical Applications
abstract

Abstract

Many applications in biology and medicine have special demands towards modeling and numerical simulation. If the blood flow in damaged vessels is considered, active material changes by rupture, material fatigue of growth must be taken into account. Similar problems arise, if biological pattern formation processes are studied. In the first part of the talk, we will introduce a numerical framework for the simulation of active materials, where the dynamics of the solid is coupled to chemical reaction systems, that can cause material alterations like growth or fatigue. We demonstrate these mechanisms on a problem of biological pattern formation, where an interplay of mechanical forces and chemical dynamics allows us to describe the formation of stable patterns. Usually such processes are described by the Turing mechanisms and involve the interplay of chemical substances. Our model is based on the interaction of a mechano-chemical system. Another problem typical in hemodynamics is the appearance of very large deformations in flow systems. Stenoses can cause full clogging of vessels, heart valves are opening and closing. Both are examples for fluid-structure interaction problems, where contact can happen. Traditional approaches, that are robust for hemodynamical fluid-structure interactions are limited to small deformations. In the second part of the talk, we will introduce a Fully Eulerian model for fluid-structure interactions, that is well suited for typical applications in biology and medicine, that require high robustness due to the added mass effect and that must deal with very large deformation up to contact.

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Wed 23/11/16 16:00
Murray Sem. Rm.
Maths/CBio/DDU/Lifespace Seminar
Dr. Paddy Brock (University of Glasgow)
The epidemiology of zoonotic malaria (Plasmodium knowlesi) in South East Asia
abstract

Abstract

Paddy Brock is a Research Associate in the Institute of Biodiversity Animal Health and Comparative Medicine at the University of Glasgow. Paddy’s research is focused on quantitative approaches to understanding disease transmission, particularly in multi-host systems that involve wildlife. He is currently working on project on the zoonotic malaria Plasmodium knowlesi, for which he is integrating analysis tools from ecology and epidemiology. Previously when based at Imperial College London he used statistical and mechanistic models to assess transmission-blocking interventions for malaria, looking at how they are influenced by parasite density. He was an author on the malaria candidate paper that the DDU published in Nature in 2015. Lifespace (http://lifespace.dundee.ac.uk) were keen to bring Paddy to Dundee to discuss his work on the Silent Signal exhibition, currently on display in LifeSpace until 26th November. Paddy collaborated with artists boredomresearch on the animation Afterglow which explores the intimate relationship between disease and its environment, using gaming engine software and mathematical modelling data. Afterglow as part of the Silent Signal exhibition is on display from 29 September - 26 November 2016.

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Mon 28/11/16 15:00
Fulton G20
Mathematics Seminar
Prof. Philippa Browning (University of Manchester)
Relaxation, heating and particle acceleration in twisted flux ropes in the solar corona
abstract

Abstract

Two major problems in astrophysics are to explain how solar coronal plasma is heated to millions of degrees – much hotter than the photospheric surface - and how energy is released and transported in solar flares. The fundamental process of magnetic reconnection, which restructures the magnetic field topology and efficiently dissipates stored magnetic energy, plays a crucial role in both solar coronal heating and flares, as well as in many other astrophysical and laboratory plasmas. Twisted magnetic flux ropes are fundamental building blocks of the solar coronal magnetic field and provide a reservoir of free magnetic energy. The talk will focus on describing mathematical models of reconnection in such structures, and the implications for solar flares, coronal heating and beyond. First, the energy release in an individual twisted magnetic flux rope will be discussed, in which magnetic reconnection is triggered in the nonlinear phase of the ideal kink instability. Using the results of 3D magnetohydrodynamic simulations coupled with a test-particle code, it will be shown how multiple magnetic reconnection sites can heat plasma and accelerate charged particles. The observational signatures of this process, such as EUV, hard X-rays and microwaves, will be discussed. Then, interacting magnetic flux ropes are considered. It will be shown how instability in a single unstable loop may trigger reconnection with stable neighbouring loops, releasing the stored energy in these loops and leading to an "avalanche" of heating events, which can be modelled both through 3D MHD simulations and relaxation theory.

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