Past

For incompressible Navier-Stokes equations in a bounded domain, I will

first present a formula for the pressure that involves the commutator

of the Laplacian and Leray-Helmholtz projection operators. This

commutator and hence the pressure is strictly dominated by the viscous

term at leading order. This leads to a well-posed and computationally

congenial unconstrained formulation for the Navier-Stokes equations.

Based on this pressure formulation, we will present a new

understanding and design principle for third-order stable projection

methods. Finally, we will discuss the delicate inf-sup stability issue

for these classes of methods. This is joint work with Bob Pego and Jie Liu.

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way.  Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.

I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold.  A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained.  This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.

All necessary tools from stochastic calculus and geometry will be explained.

Abstract:  Modern techniques for computing multi-particle and multi-loop scattering amplitudes rely on a sophisticated use of on-shell recursion relations and generalised unitarity methods. I will show that these methods are ideally suited to interpretation in twistor space, where superconformal properties become manifest. In fact, the recursion relations of Britto, Cachazo, Feng & Witten provide a clear framework for the twistor diagram program initiated in the 1970s.
Tree-level scattering amplitudes in N=4 SYM are now known to possess a Yangian symmetry, formed by combining the original PSU(2,2|4) superconformal invariance with a second "dual" copy. I will also discuss very recent work constructing scattering amplitudes in a twistor space in which this dual superconformal symmetry acts geometrically.

Soliton like structures called “stable droplets” are found to exist within a paradigm reaction
diffusion model which can be used to describe the patterning in a number of fish species. It is
straightforward to analyse this phenomenon in the case when two non-zero stable steady states are
symmetric, however the asymmetric case is more challenging. We use a recently developed
perturbation technique to investigate the weakly asymmetric case.

We focus on structural models in corporate finance with roll-over debt structure and endogenous default triggered by limited liability equity-holders. We point out imprecisions and misstatements in the literature and provide a rationale for the endogenous default policy.

Inverse problems arise with regularity (sic!) in the context of our study of the deformation of solids, and its characterisation (in terms of diffraction and imaging) using radiation (neutrons and X-rays).

I wish to introduce several examples where the advancement of inverse problem methods can make a significant impact on applicatins.

1. Inverse eigenstrain analysis of residual stress states

2. Strain tomography

3. Strain image correlation

Depending on the time available, I may also mention (a) Rietveld refinement of diffraction patterns from polycrystalline aggregates, and
(b) Laue pattern indexing and energy dispersive detection for single grain strain analysis.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.

This is an overview, mostly of work of others (Denef, Loeser, Merle, Heinloth-Bittner,..). In the first part of the talk we give a brief introduction to motivic integration emphasizing its application to vanishing cycles. In the second part we discuss a join construction and formulate the relevant Sebastiani-Thom theorem.

We propose a powerful prediction algorithm built upon Gaussian
processes (GPs). They are particularly useful for their flexibility,
facilitating accurate prediction even in the absence of strong physical models. GPs further allow us to work within a completely Bayesian framework. As such, we show how the hyperparameters of our system can be marginalised by use of Bayesian Monte Carlo, a principled method of approximate integration. We employ the error bars of the GP's prediction as a means to select only the most informative observations to store. This allows us to introduce an iterative formulation of the GP to give a dynamic, on-line algorithm. We also show how our error bars can be used to perform active data selection, allowing the GP to select where and when it should next take a measurement.

We demonstrate how our methods can be applied to multi-sensor prediction problems where data may be missing, delayed and/or correlated. In particular, we present a real network of weather sensors as a testbed for our algorithm.

The study of representation growth of infinite groups asks how the

numbers of (suitable equivalence classes of) irreducible n-dimensional

representations of a given group behave as n tends to infinity. Recent

works in this young subject area have exhibited interesting arithmetic

and analytical properties of these sequences, often in the context of

semi-simple arithmetic groups.

In my talk I will present results on the representation growth of some

classes of finitely generated nilpotent groups. They draw on methods

from the theory of zeta functions of groups, the (Kirillov-Howe)

coadjoint orbit formalism for nilpotent groups, and the combinatorics

of (finite) Coxeter groups.

The Kerr solutions to Einstein's equations describe rotating black holes. For the wave equation in flat-space and outside the non-rotating, Schwarzschild black holes, one method for proving decay is the vector-field method, which uses the energy-momentum tensor and vector-fields. Outside the Schwarzschild black hole, a key intermediate step in proving decay involved proving a Morawetz estimate using a vector-field which pointed away from the photon sphere, where null geodesics orbit the black hole. Outside the Kerr black hole, the photon orbits have a more complicated structure. By using the hidden symmetry of Kerr, we can replace the Morawetz vector-field by a fifth-order operator which, in an appropriate sense, points away from the photon orbits. This allows us to prove the necessary Morawetz estimate. From this we can prove a decay estimate of almost $t^{-1}$ for fixed $r$ and the corresponding decay rates at the event horizon and null infinity. The major innovation in this result is that, by using the hidden symmetries with the energy-momentum, we can avoid taking Fourier tranforms in time.

This is joint work with Lars Andersson.