Past

Stability is central to the study of numerical algorithms for solving
partial differential equations. But stability can be subtle and elusive. In
fact, for a number of important classes of PDE problems, no one has yet
succeeded in devising stable numerical methods. In developing our
understanding of stability and instability, a wide range of mathematical
ideas--with origins as diverse as functional analysis,differential geometry,
and algebraic topology--have been enlisted and developed. The talk will
explore the concept of stability of discretizations to PDE, its significance,
and recent advances in its understanding.

While the average settling velocity of particles in a suspension has been successfully predicted, we are still unsuccessful with the r.m.s velocity, with theories suggesting a divergence with the size of

the container and experiments finding no such dependence. A possible resolution involves stratification originating from the spreading of the front between the clear liquid above and the suspension below. One theory describes the spreading front by a nonlinear diffusion equation

$\frac{\partial \phi}{\partial t} = D \frac{\partial }{\partial z}(\phi^{4/5}(\frac{\partial \phi}{\partial z})^{2/5})$.

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Experiments and computer simulations find differently.

Using a technique explored in unpublished work of Ball and Mizel I shall

show that already in 2 and 3 dimensions there are vectorfields which are

singular minimizers of integral functionals whose integrand is strictly

polyconvex and depends on the gradient of the map only. The analysis behind

these results gives rise to an interesting question about the relationship

between the regularity of a polyconvex function and that of its possible

convex representatives. I shall indicate why this question is interesting in

the context of the regularity results above and I shall answer it in certain

cases.

Consider a graph with n vertices placed randomly in the unit

square, each connected by an edge to its nearest neighbour in a

south-westerly direction. For many graphs of this type, the centred

total length is asymptotically normal for n large, but in the

present case the limit distribution is not normal, being defined in

terms of fixed-point distributions of a type seen more commonly in

the analysis of algorithms. We discuss related results. This is

joint work with Andrew Wade.

We consider a system of interacting Fisher-Wright diffusions

which arise in population genetics as the diffusion limit of a spatial

particle model in which frequencies of genetic types are changing due to

migration and reproduction.

For both models the historical processes are constructed,

which record the family structure and the paths of descent through space.

For any fixed time, particle representations for the

historical process of a collection of Moran models with increasing particle

intensity and of the limiting interacting Fisher-Wright diffusions are

provided on one and the same probability space by means of Donnelly and

Kurtz's look-down construction.

It will be discussed how this can be used to obtain new

results on the long term behaviour. In particular, we give representations for

the equilibrium historical processes. Based on the latter the behaviour of

large finite systems in comparison with the infinite system is described on

the level of the historical processes.

The talk is based on joint work with Andreas Greven and Vlada

Limic.

We consider the problem of recovering the position of a scattering surface

from measurements of the scattered field on a finite line above the surface.

A point source algorithm is proposed, based on earlier work by Potthast,

which reconstructs, in the first instance, the scattered field in the whole

region above the scattering surface. This information is used in a second

stage to locate the scatterer. We summarise the theoretical results that can

be obtained (error bounds on the reconstructed field as a function of the

noise level in the original measurements). For the case of a point source of

the incident field we present numerical experiments for both a steady source

(time harmonic excitation) and a pulse source typical of an antenna in

ground penetrating radar applications.

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This is joint work with Claire Lines (Brunel University).

After a review of the symmetries of supergravity theories and Kac-Moody

algebras, we explain show that M theory is likely to possess a very large

Kac-Moody symmetry of rank eleven, denoted by E_{11}. We also provide

evidence that even pure gravity and the closed bosonic string possess

analogous Kac-Moody symmetries.

Finally, we explain how the central charges of the maximal supergravity

theories arise naturally in E_{11}.

We consider the Cahn Hilliard Equation in the line, perturbed by

the space derivative of a space--time white noise. We study the

solution of the equation when the initial condition is the

interface, in the limit as the intensity of the noise goes to zero

and the time goes to infinity conveniently, and show that in a scale

that is still infinitesimal, the solution remains close to the

interface, and the fluctuations are described by a non Markovian

self similar Gaussian process whose covariance is computed.

After a brief introduction to the basics of Rough Paths I'll

explain recent work by Peter Friz, Dan Stroock and myself proving that a

Brownian path conditioned to be uniformly close to a given smooth path

converges in distribution to that path in the Rough Path metric. The Stroock

Varadhan support theorem is an immediate consequence.

The novel part of the argument is to

obtain the estimate in a way that is independent of the particular norm used

in the Euclidean space when one defines the uniform norm on path space.

The seminar will address issues related to numerical simulation

of non-linear behaviour of solid materials to impact loading.

The kinematic and constitutive aspects of the transition from

continuum to discontinuum will be presented as utilised

within an explicit finite element development framework.

Material softening, mesh sensitivity and regularisation of

solutions will be discussed.