Term 2 2011-2012
List of Speakers:
Generating pseudorandom numbers
It is impossible for deterministic computers to generate truly random samples for stochastic simulations. The aim of a pseudorandom number generator is to generate numbers that are indistinguishable from truly random samples in the sense that they are not rejected by hypothesis tests. I will give examples of pseudorandom generators (good and bad) and explain some important properties and theorems associated with the construction of a realistic and efficient generator. Elementary number theory will make a surprise(?) appearance!
A finite element method for solving partial differential equations on evolving surfaces
In 1988, Gerhard Dziuk introduced the surface finite element method to solve the Poisson equation on arbitrary surfaces. Here, in this talk I will present recent work in which we derive a Cahn-Hilliard type equation on a given evolving surface. Questions of existence and uniqueness are addressed followed by a convergence result assuming regularity of solutions. If time allows I will then show how the method can be used to solve equations where the evolving of the surface is also an unknown. In particular, a natural time discretisation is used which gives better stability properties than other current methods.
Random Spatial Permutations and the Existence of Bose-Einstein Condensation
In the 1920s Bose and Einstein conjectured a phase transition for gasses cooled to very low temperatures. 70 years later scientists were able to physically show the existence of this phenomenon, now known as Bose Einstein condensation (BEC), and were duly awarded Nobel prizes. In the 1950s Feynman suggested a mathematical model which he believed to capture the notion of BEC, however whilst the physical phase transition has been seen, as of yet no proof has been given of phase transition for the mathematical model. We provide an overview of Feynman's formulation, and discuss its links to the theory of random spatial permutations.
Variational discretization of gradient flows
Various interesting evolution problems can be formulated as gradient flows. We can use the variational nature of these flows to derive a family of time-discrete numerical schemes where an optimization program is solved at every time step. We discuss applications for 4th order PDEs on surfaces and for geometric flows.
Term 1 2011-2012
List of Speakers:
Conformally Invariant Scaling Limits
There are many mathematical models of statistical physics in 2 dimensions that are believed to benefit from conformal invariance. These predictions have foundations in physics but have more recently become accessible to mathematical study. In particular, Smirnov (2007) proved the asymptotic conformal invariance of a specific system: critical site percolation on the planar triangular lattice. Smirnov's proof can be used to motivate the definition of Stochastic Loewner evolutions (SLE). SLE appear as limits of interfaces or paths occurring in a variety of statistical mechanics models as the mesh of the grid on which the model is defined tends to zero.
Importance sampling on genetic ancestries modelled with the Kingman coalescent
Given a sample of genetic information for some set of individuals, population genetics is used to infer the ancestries most likely to correspond to the present-day population. Of particular interest are the mutation rate and the time to the most recent common ancestor (TMRCA) of the sample. In this talk I explain how the Kingman coalescent model arises as the limiting case of a certain set of exchangeable population genetics models and demonstrate its use in an importance sampling procedure for estimating the mutation rate.
Stochastic Partial Differential Equations on Evolving Riemannian Manifolds
The theory of SPDEs on evolving manifolds is devoid in the mathematical literature, at odds with the deterministic counter-part. The aim of this talk is to introduce the theory of SPDEs on evolving Riemannian manifolds, in the variational setting to the analysis of SPDEs. I will motivate the topic, look at the general theory of SPDEs in the variational setting before producing the theory in the simple case. Don't worry, there will be examples!
Discontinuous Galerkin Methods for Surface Partial Differential Equations
Partial differential equations on manifolds have become an active area of research in recent years due to the fact that in many applications, problems do not reside on a flat Euclidean domain but on a curved hypersurface. Exact solutions for such problems are rare, so we need to resort to numerical methods as an approximation. Finite element methods (FEM) have been successfully extended to surfaces from both a theoretical and numerical point of view. However, it is well-known that there are a number of situations where FEM may not be the appropriate numerical method and there has been comparatively little done to investigate alternative numerical methods. The talk aims to outline work done on extending the Discontinuous Galerkin (DG) framework onto hypersurfaces. A priori error estimates are then verified numerically for simple test problems.
Stability Analysis of The Atomistic and QC Approximations for The EAM Model
The accurate approximation of critical strains for lattice instability is a key criterion for predictive computational modeling of materials. We will present a comparison of the lattice stability for atomistic chains modeled by the embedded atom method (EAM) with their approximation by local Cauchy-Born models. We find that both the volume-based local model and the reconstruction-based local model can give O(1) errors for the critical strain since the embedding energy density is generally strictly convex. The critical strain predicted by the volume-based model is always larger than that predicted by the atomistic model, but the critical strain for reconstruction-based models can be either larger or smaller than that predicted by the atomistic model.
Stationary Euler flows and ideal magnetohydrodynamics
The Euler equations form one of the simplest possible models for fluid flow, in which the viscosity is neglected; yet, in three dimensions, there is no known global existence and uniqueness result. In this talk we consider stationary solutions of the Euler equations as a means of studying their long-time behaviour. In particular, we consider the "analogy" proposed by Moffatt (1985) between stationary Euler flows and stationary magnetic fields evolving under the ideal magnetohydrodynamics equations, and show that not all is as easy as it first seems.
Today fractional calculus is a quickly delevoping area in mathematical research, which has numerous and useful applications. For example, it appears to describe anomalous diffusion, which models behaviour of electrons in semiconductors in laser printers. I will introduce you to the basics of fractional dynamics and the theory behind the processes which enjoy their friendship with fractional evolution equations.
Allen Cahn equation on surfaces
The Allen Cahn equation, introduced by Allen & Cahn (1979) to model the motion of phase boundaries in solids, has the property to approximate mean curvature flow and is a core component of phase-field models. The aim of this talk is shift the equation from a planar domain to a smooth surface in 3D. Using the method of Chen (1992) and Alfaro et al. (2009) we will derive the behaviour for the surface Allen Cahn equation and show that it can be used to approximate geodesic curvature flow.
A phase field approach to image reconstruction
We will talk about the inverse problem of reconstructing a piecewise constant function from blurred and noisy data. This has many applications, such as reading a barcode or finding the boundary between different types of tissue in a medical image. We will formulate the problem in a PDE setting using a phase field approach. We then discuss and demonstrate numerical methods for solving it.
I will explain what it means for a phase transition to occur in a thermodynamic system, and provide an example demonstrating what happens mathematically. The example is the Ising/Potts model. To show the existence of a phase transition in this model I will introduce the concept of a Gibbs measure and the FK representation (the random-cluster model). The audience will learn what a phase transition is and how the existence of a phase transition in a system is related to percolation and the existence of multiple Gibbs measures.
The Lamperti Transformation
The subject of Levy processes is one that has been greatly researched over the past century yielding vast quantities of literature. However, the fruits of this literature are often not only concerned with the Levy processes themselves, but also with the understanding they can provide in the fields of other stochastic processes. The Lamperti Transform provides just this: a bijection between Levy Processes and the, lesser-known but often more useful, positive self-similar Markov processes that can "carry across" certain properties.
Pharmacokinetic modelling of derivatives of the anti-malarial drug artemisinin
We motivate the need to generate and validate mathematical models for artemisinins and discuss the modelling process, a key feature of which is the (often overlooked) structural identifiability analysis.
Along the way, we introduce compartmental models, a class of models well-suited to pharmacokinetics. We then discuss the shortcomings of existing models before presenting a new model, constraining it to be structurally identifiable, and showing representative fits to data.