Past

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.

We prove that when a sequence of Lévy processes $X(n)$ or a normed sequence of random walks $S(n)$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L(n)$ of local times at the supremum of $X(n)$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and

descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S(n)$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.

Defaults in a credit portfolio of many obligors or in an economy populated with firms tend to occur in waves. This may simply reflect their sharing of common risk factors and/or manifest their systemic linkages via credit chains. One popular approach to characterizing defaults in a large pool of obligors is the Poisson intensity model coupled with stochastic covariates, or the Cox process for short. A constraining feature of such models is that defaults of different obligors are independent events after conditioning on the covariates, which makes them ill-suited for modeling clustered defaults. Although individual default intensities under such models can be high and correlated via the stochastic covariates, joint default rates will always be zero, because the joint default probabilities are in the order of the length of time squared or higher. In this paper, we develop a hierarchical intensity model with three layers of shocks -- common, group-specific and individual. When a common (or group-specific) shock occurs, all obligors (or group members) face individual default probabilities, determining whether they actually default. The joint default rates under this hierarchical structure can be high, and thus the model better captures clustered defaults. This hierarchical intensity model can be estimated using the maximum likelihood principle. A default signature plot is invented to complement the typical power curve analysis in default prediction. We implement the new model on the US corporate bankruptcy data and find it far superior to the standard intensity model both in terms of the likelihood ratio test and default signature plot.

We consider the three dimensional Navier-Stokes equations with a large initial data and

we prove the existence of a global smooth solution. The main feature of the initial data

is that it varies slowly in the vertical direction and has a norm which blows up as the

small parameter goes to zero. In the language of geometrical optics, this type of

initial data can be seen as the ``ill prepared" case. Using analytical-type estimates

and the special structure of the nonlinear term of the equation we obtain the existence

of a global smooth solution generated by this large initial data. This talk is based on a

work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z.

Zhang.