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2015/16 semester 1 events in Mathematics

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Mon 14/09/15 14:00
Fulton G20
Mathematics Seminar
Dr. Anthony Yeates (Durham University)
Data-driven modelling of the Sun's magnetic field
abstract

Abstract

I will discuss issues in trying to reconstruct the magnetic field structure in the solar atmosphere based on limited magnetic observations on the solar surface. This effort is important not only for understanding the Sun itself but also for modelling and predicting space weather. The crux of the problem is inverting Faraday's law to determine the electric field from observations of only the magnetic field.

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Mon 21/09/15 14:00
Fulton G20
Mathematics Seminar
Dr. Rastko Sknepnek (University of Dundee)
Flocks on a sphere: Effects of curvature on active collective motion
abstract

Abstract

In this talk we show that coupling to curvature has profound effects on collective motion in active systems, leading to patterns not observed in flat space. Biological examples of such active motion in curved environments are numerous: curvature and tissue folding are crucial during gastrulation, epithelial and endothelial cells move on constantly growing, curved crypts and vili in the gut, and the mammalian corneal epithelium grows in a steady-state vortex pattern. Using Brownian dynamics simulations, we study a model of self-propelled particles with polar alignment on a sphere. Hallmarks of these motion patterns are a polar vortex and a circulating band arising due to the incompatibility between spherical topology and uniform motion - a consequence of the hairy ball theorem. Furthermore, inspired by recent experiments of Baush and Dogic groups on droplets coated with actively driven microtubule bundles, we address active nematic systems confined to the surface of a sphere.

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Tue 29/09/15 14:00
Baxter Suite
1st Floor Tower
Careers Advisors (Mathematics Careers Day)
University of Dundee
Tue 06/10/15 14:00
Fulton G20
Mathematics Seminar
Prof. Charalambos Makridakis (University of Sussex)
Self adaptive computational methods for nonlinear phenomena
abstract

Abstract

The computation of singular phenomena (shocks, defects, dislocations, interfaces, cracks) arises in many complex systems. For computing such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. At the same time, we would like to have mathematical guarantees that our computational methods approximate physically relevant solutions. Our purpose in this talk is to review results and discuss related computational challenges for such nonlinear problems modeled by PDEs. In addition we shall discuss issues related to Micro / Macro adaptive modeling and methods, in particular related to atomistic/continuum coupling in crystalline materials.

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Mon 12/10/15 14:00
Fulton G20
Mathematics Seminar
Dr. Andrew Hillier (DAMTP - Cambridge University)
Prominences, the magnetic Rayleigh-Taylor instability and what we can learn about the prominence magnetic field
abstract

Abstract

Observations by the Hinode satellite of quiescent prominences have revealed in great detail the dynamics of plumes rising through the prominence material. These plumes, created by the magnetic Rayleigh-Taylor instability, rise through the prominence material. In this talk I will show how the growth rate for the linear instability and the nonlinear dynamics of the rising plume in the nonlinear regime provide diagnostic tools for investigating both the plasma beta and the magnetic field direction in the prominence. These methods will be compared to observations of plumes with both Doppler velocity and magnetic field measurements (both strength and direction) by Orozco Suarez et al (2014) to confirm their validity. It is through application of these new methods that I believe we will be able to make great strides in understanding the role of the magnetic field in the small-scale dynamics of quiescent prominences without the need for complex polarization measurements.

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Mon 19/10/15 14:00
Fulton G20
Mathematics Seminar
Dr. Omar Lakkis (University of Sussex)
PDEs of Monge-Ampère type and their numerical approximation
abstract

Abstract

This talk is directed to a general audience, I will thus spend some time on the historical and mathematical background of the Monge-Ampère partial differential equation and its relation to optimal mass transportation and geometric problems (including classical geometric optics as a surprising link between the two disciplines). In a second, more technical part I will brief on the various numerical methods, with a focus on the Nonvariational Finite Element Method, developed as a joint work with Tristan Pryer (Reading) and Ellya Kawecki (Oxford).

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Mon 26/10/15 14:00
Fulton G20
Mathematics Seminar
Prof. Hans Othmer (University of Minnesota)
Analysis of models for direction sensing, actin waves, and surface deformations in motile cells
abstract

Abstract

Cell locomotion is essential for early development, angiogenesis, tissue regeneration, the immune response, and wound healing in multicellular organisms, and plays a very deleterious role in cancer metastasis in humans. Locomotion involves the detection and transduction of extracellular chemical and mechanical signals, integration of the signals into an intracellular signal, and the spatio-temporal control of the intracellular biochemical and mechanical responses that lead to force generation, morphological changes and directed movement. In this talk we will discuss mathematical models of direction sensing, how actin waves can be generated in cells, and how a cell can propel itself via surface deformations.

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Mon 02/11/15 14:00
Fulton G20
Mathematics Seminar
Dr. Radostin Simitev (University of Glasgow)
Convection-driven dynamos in spherical shells and applications to the Solar Dynamo
abstract

Abstract

Simple self-consistent models of magnetic field generation by convection in rotating spherical shells exhibit properties resembling those observed on the Sun. I will provide a review of some basic phenomenology of finite-amplitude convection and dynamo action in rotating spherical shells, including dependence on dimensionless parameters and dipolar dynamo bistability. I will then describe our recent efforts to apply these results to mimic properties of the Solar differential rotation, equatorial-ward migration of sunspots and active longitudes. I will also comment on the dynamo effects near the transition from Solar to antisolar differential rotation.

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Mon 09/11/15 14:00
Fulton G20
Mathematics Seminar
Prof. Jeremy Levesley (University of Leicester)
Sparse grid approximation with Gaussians
abstract

Abstract

Sparse grid approximation with polynomial splines has been a very successful method for approximation and solving PDEs. A downside to having to make a choice of degree of spline is that the convergence order is limited by this choice. We are developing sparse grid algorithms using radial basis functions, in particular the Gaussian. However, the Gaussian has the strange property that in a stationary scheme there is no convergence. So a multilevel method is required. We show numerically that this has good convergence properties. We will also present preliminary results that give convergence rates for multilevel quasi-interpolation. This result is the first multilevel result for smooth basis functions, and will open the door for sparse grid convergence estimates more generally (we hope!).

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Mon 16/11/15 14:00
Fulton G20
Mathematics Seminar
Prof. Hua Chen (Wuhan University)
Estimates of Dirichlet Eigenvalues for Degenerate Elliptic Operators
abstract

Abstract

Let $\Omega$ be a bounded open domain in $R^n$ with smooth boundary and $X=(X_1, X_2, \cdots, X_m)$ be a system of real smooth vector fields defined on $\Omega$ with the boundary $\partial\Omega$ which is non-characteristic for $X$. If $X$ satisfies the Hormander's condition, then the vector fields is finite degenerate and the sum of square operator $\triangle_{X}=\sum_{j=1}^{m}X_j^2$ is a finitely degenerate elliptic operator, otherwise the operator $-\triangle_{X}$ is called infinitely degenerate. If $\lambda_j$ is the $j^{th}$ Dirichlet eigenvalue for $-\triangle_{X}$ on $\Omega$, then this paper shall study the lower bound estimates for $\lambda_j$. Firstly, by using the sub-elliptic estimate directly, we shall give a simple lower bound estimates of $\lambda_j$ for general finitely degenerate $\triangle_{X}$ which is polynomial increasing in $j$. Secondly, if $\triangle_{X}$ is so-called Grushin type degenerate elliptic operator, then we can give a precise lower bound estimates for $\lambda_j$. Finally, by using logarithmic regularity estimate, for infinitely degenerate elliptic operator $\triangle_{X}$ we prove that the lower bound estimates of $\lambda_j$ will be logarithmic increasing in $j$.

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Mon 23/11/15 14:00
Fulton G20
Mathematics Seminar
Dr. Nikos Kavallaris (University of Chester)
On a non-local parabolic problem arising in game theory
abstract

Abstract

In the current talk we first present the construction of a degenerate non-local equation which arises in the replicator dynamics system coming from the evolutionary game theory. Once the local existence is established we study the long-time behaviour of the preceding equation. Depending on the total mass of the initial data we either prove global-time existence or finite-time blow-up. For total mass equal to 1 then also the convergence towards the steady-states (Nash equilibria) is proven

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Mon 30/11/15 14:00
Fulton G20
Mathematics Seminar
Mr. Christopher Berg Smiet (University of Leiden)
Hopf Fibrations and Field Lines
abstract

Abstract

The Hopf map is a curious result in algebraic topology which was given in 1931 by Heinz Hopf. He showed a map between the hypershpere S^3 and the sphere S^2, in such a way that the fibers of the map (pre-images of points on the sphere) are all circles that are all linked with one another. This was a breakthrough in topology, but in applied mathematics this result has lain dormant for many years. Recently there has been a revived interest in topological aspects of physical systems, and the topological structure the Hopf map has been applied to describe many physical systems. In this talk I will give a brief overview of the Hopf map, and how it can be applied as a versatile tool to construct new solutions of Maxwell's equations, describe a inherently self-stable plasma configuration, and be adapted to approximate the field of a relaxing helical plasma.

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