Past

Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces

for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients.

The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator

and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO)

which guarantee that the solution operator associated with this problem is an isomorphism.

Hairsine-Rose (HR) model is the only multi sediment size soil erosion

model. The HR model is modifed by considering the effects of sediment bedload and

bed elevation. A two step composite Liska-Wendroff scheme (LwLf4) which

designed for solving the Shallow Water Equations is employed for solving the

modifed Hairsine-Rose model. The numerical approximations of LwLf4 are

compared with an independent MOL solution to test its validation. They

are also compared against a steady state analytical solution and experiment

data. Buffer strip is an effective way to reduce sediment transportation for

certain region. Modifed HR model is employed for solving a particular buffer

strip problem. The numerical approximations of buffer strip are compared

with some experiment data which shows good matches.

We reconsider Merton's problem under proportional transaction costs.

Beginning with Davis and Norman (1990) such utility maximization problems are usually solved using stochastic control theory.

Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process.

In this study we show that this dual approach can actually be used for both deriving a candidate solution and verification.

In particular, the shadow price process is determined explicitly.

Collaborators from Industry will speak to us about their proposed projects for the MSc in Math Modelling and Scientific Computation. Potential supervisors should attend. All others welcome too.

After a short introduction to the section conjecture, I plan to present a "minimalistic" form of the birational p-adic section conjecture. The result is related to both: Koenigsmann's proof of the birational p-adic section conjecture, and a "minimalistic" Galois characterisation of formally p-adic valuations.

Whispering gallery modes in optical resonators have received a lot of attention as a mechanism for constructing small, directional lasers. They are also potentially important as passive optical components in schemes for coupling and filtering signals in optical fibres, in sensing devices and in other applications. In this talk it is argued that the evanescent field outside resonators that are very slightly deformed from circular or spherical is surprising in a couple of respects. First, even very small deformations seem to be capable of leading to highly directional emission patterns. Second, even though the undelying ray families are very regular and hardly differ from the integrable circular or spherical limit inside the resonator, a calculation of the evanescent field outside it is not straightforward.

This is because even very slight nonintegrability has a profound effect on the complexified ray families which guide the external wave to asymptopia. An approach to describing the emitted wave is described which is based on canonical perturbation theory applied to the ray families and extended to comeplx phase space.

Molecular Dynamics Simulations are a tool to study the behaviour

of atomic-scale systems. The simulations themselves solve the

equations of motion for hundreds to millions of particles over

thousands to billions of time steps. Due to the size of the

problems studied, such simulations are usually carried out on

large clusters or special-purpose hardware.

At a first glance, there is nothing much of interest for a

Numerical Analyst: the equations of motion are simple, the

integrators are of low order and the computational aspects seem

to focus on hardware or ever larger and faster computer

clusters.

The field, however, having been ploughed mainly by domain

scientists (e.g. Chemists, Biologists, Material Scientists) and

a few Computer Scientists, is a goldmine for interesting

computational problems which have been solved either badly or

not at all. These problems, although domain specific, require

sufficient mathematical and computational skill to make finding

a good solution potentially interesting for Numerical Analysts.

The proper solution of such problems can result in speed-ups

beyond what can be achieved by pushing the envelope on Moore's

Law.

In this talk I will present three examples where problems

interesting to Numerical Analysts arise. For the first two

problems, Constraint Resolution Algorithms and Interpolated

Potential Functions, I will present some of my own results. For

the third problem, using interpolations to efficiently compute

long-range potentials, I will only present some observations and

ideas, as this will be the main focus of my research in Oxford

and therefore no results are available yet.

According to the Ernst-Geroch reduction, in an axially symmetric stationary electrovac spacetime, the Einstein-Maxwell equations reduce to a harmonic map problem with singular boundary data. I will discuss the “regularity” of the reduced harmonic maps near the boundary and its implication on the regularity of the corresponding spacetimes.

We describe John Stalling's method of studying finitely generated free groups via graphs and moves on graphs called folds. We will then discuss how the theory can be extended to study the automorphism group of a finitely generated free group.

The representation theory of groups is surrounded by deep and difficult conjectures. In this talk we will take a tour of (some of) these problems, including Alperin's weight conjecture, Broué's conjecture, and Puig's finiteness conjecture.

A classic problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated commutative algebra by algebra automorphisms, whether the ring of invariants is still finitely generated. I shall present joint work with W. van der Kallen treating the problem for a Chevalley group over an arbitrary base. Progress on the corresponding problem of finite generation for rational cohomology will be discussed.

Algebraic classification of the Weyl tensor is an important tool for solving the Einstein equation. I shall review the classification for spacetimes of dimension greater than four, and recent progress in using it to construct new exact solutions. The higher-dimensional generalization of the Goldberg-Sachs theorem will be discussed.

The talk starts with the observation that many well-known systems of diffusive type

can be written as Wasserstein gradient flows. The aim of the talk is

to understand _why_ this is the case. We give an answer that uses a

connection between diffusive PDE systems and systems of Brownian

particles, and we show how the Wasserstein metric arises in this

context. This is joint work with Johannes Zimmer, Nicolas Dirr, and Stefan Adams.