# 2012-2013

**Term 2 2012-2013**

**List of Speakers:**

13/03/13

**Unfitted finite element methods for surface partial differential equations**

**Tom Ranner**

Surface partial differential equations have grown in popularity within the last twenty years with applications in fluid mechanics, biology and material sciences becoming increasingly common. A traditional approach is to triangulate the surface and apply a finite element method based on the piecewise linear surface. These methods rely on being able to generate an accurate, regular triangulation which in practice may be difficult to find. In this talk, we propose an alternative approach using a bulk finite element space to solve an embedded version of the surface equations. We apply this methodology to derive two methods for solving a Poisson equation and go on to show how one may use similar methods to solve a heat equation on an evolving surface.

23/01/13

**Modelling surfactants in two-phase flow**

**Andrew Lam**

I will derive a classical model of surfactants in a two-fluid system using a sharp interface description (where the interface between the two fluids has zero width). However, this description breaks down when we observe topological transitions: i.e. when surfaces collide, when the interface self-intersects or pinches off. Since these transitions occur naturally, I will derive an alternative description based on phase field models, where the sharp interface is replaced by an interfacial layer of finite width. At the core of these models is an order parameter that is used to distinguish the fluids and interface.

I will show that the sharp interface model can be recovered from the phase field model, as well as some numerical results in 1D and 2D

16/01/13

**Spatial Modelling and Simulation of Ran-driven Nuclear Transport**

**Maha Alhajri**

We consider the Ran-driven nuclear transport in the living cell, and analyse the reaction network behind the transport phenomena through the nuclear envelop. Then we present a mathematical model of reaction diffusion system equipped with interface conditions imitating the nuclear envelop. The model is then constructed by means of mass action and enzyme kinetics.

Then we introduce discontinuous Galerkin methods as a logical numerical method to treat the interface conditions. By first justifying the weak discontinuous Galerkin form, we make use of it in DUNE package to perform numerical simulations and experiments. We also provide an elementary

numerical scheme stability analysis of the fully discrete model. The numerical treatment is obtained by constructing two schemes Imp/Exp and

fully implicit where we hope that leads to a better numerical scheme for the treatment of the model in 3D dimension.

**Term 1 2012-2013**

**List of Speakers:**

5/12/12

**Matthew Thorpe**

**Multitarget Tracking in a GSM Network**

Using three sensors at different locations it is possible to observe the signal from a mobile phone and calculate its position based on differences in time of arrival. The signal contains information such as frequency channel and a synchronisation code that can be used to match signals to phones. Errors tend to be very high due to signals travelling at the speed of light. Even a small error in the time of arrival can lead to a very large difference in location. By correcting for bias we have been able to get more realistic results. This talk gives an introduction to the GSM network and presents results from the first Warwick experiment carried out over summer.

**Cell Motility**

**Amal Alphonse**

This talk will cover some of the material in my MSc thesis in which I considered the well-posedness of certain partial differential equations (PDEs) that arise from modelling cell motility. Namely, these equations include a surface PDE that models a quantity present on the surface of the cell and a PDE coming out of the mean curvature flow that specifies the evolution of the cell. We first give an existence result in the space of parabolic Holder functions for a general class of nonlinear parabolic PDEs by means of a linearisation method, and then we apply this theory to cell motility.

28/11/12

**Markov Functions**

**YuChen Pei**

When does a function f map a Markov process (X_t) to a Markov process (f(X_t))? I'll discuss, compare and give examples of two cases: Dynkin's criterion and Pitman-Rogers' theorem.

**Data assimilation scheme**

**Abhishek Shukla**

In this talk we discuss few Data assimilation schemes. It is of particular interest for non-linear dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion. We also discuss results for 3DVAR scheme with particular example of Lorenz'63 model."

31/10/12

**Dyson's Brownian motion**

**Chin Lun**

Consider a Hermitian matrix whose entries are independent Brownian motions. We will show that the N eigenvalues of such a matrix is a diffusion called Dyson's Brownian motion. We derive the SDE which the eigenvalues satisfy and show that Dyson's Brownian motion can also be realised as the h-transform of a certain process.

**Schramm–Loewner evolution**

**Alex Kister**

This talk is a overview over Schramm–Loewner evolution. At first the derivation via Loewner evolution is outlined. Then two characteristic properties of the Schramm-Loewner evolution are introduced, this two properties could also be used to define the evolution. Finally the connection between loop erased random walks and the Schramm–Loewner evolution is cited.

24/10/12

**The Random Interlacement Model**

**Owen Daniel**

We provide a gentle introduction to the model of random interlacements, an example of a highly correlated percolation model on an infinite graph. The model was introduced by Sznitman, who considered the trajectory of a random walk on a large torus for : at a local level, the image of the random interlacement model on a small box in looks similar to the trajectory of the random walk in the same box in .

We will describe this relationship, in particular highlighting its relationship with the cover time of the torus, a recent result due to Belius.

**The Infinitesimal Pairwise Immigration Model**

**Barnaby Garrod**

The Infinitesimal Pairwise Immigration (IPI) model is a system of interacting 1-dimensional Brownian motions, in which pairs of particles are immigrated as time evolves and particles instantaneously annihilate upon meeting. We motivate the model by relating it to kink-antikink pairs of an SPDE, and provide an outline of its construction as a limit of continuous processes. Along the way, we meet the Brownian Web and its dual, which enable us to obtain formulae characterising the model.

17/10/12

**Density Functional Theory: The Classical Hard-Core Gas**

**Simon Bignold**

This talk will cover some of the material used for my MSc dissertation, supervised by Christoph Ortner. The topic of the dissertation was density functional theory, specifically a classical consideration of the one-dimensional hard-core gas in the canonical ensemble.

The talk begins by introducing some basic techniques required for density functional theory, after which we develop the ideal gas as a motivating example. We conclude the talk with a brief look at one of the methods used to consider the one-dimensional hard-core gas.

Density functional theory is a topic of interest in mathematics, physics and in chemistry. This talk reviews material required for a specific problem in the hope of demonstrating some of the techniques and problems encountered in density functional theory. This talk should hopefully be accessible to anyone with a reasonable degree of mathematical literacy.