Past

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.

This work in collaboration with J. Casado-Díaz deals with the asymptotic behaviour of two-dimensional linear conduction problems for which the sequence of conductivity matrices is bounded from below but not necessarily from above. On the one hand, we prove an extension in dimension two of the classical div-curl lemma, which allows us to derive a H-convergence type result for any L¹-bounded sequence of conductivity matrices. On the other hand, we obtain a uniform convergence result satisfied by the minimisers of a sequence of two-dimensional diffusion energies. This implies the closure for the L²-strong topology of $\Gamma$-convergence of the sets of equicoercive diffusion energies without assuming any bound from above. A few counter-examples in dimension three, connected with the appearance of non-local effects, show the specificity of dimension two in the two previous compactness results.

Can investors with irrational beliefs be neglected as long as they are rational on average ? Does unbiased disagreement lead to trades that cancel out with no consequences on prices, as implicitly assumed by the traditional models ? We show in this paper that there is an important impact of unbiased disagreement on the behavior of financial markets, even though the pricing formulas are "on average" (over the states of the world) unchanged. In particular we obtain time varying, mean reverting and countercyclical (instead of constant in the standard model) market prices of risk, mean reverting and procyclical (instead of constant) risk free rates, decreasing (instead of flat) yield curves in the long run, possibly higher returns and higher risk premia in the long run (instead of a flat structure), momentum in stock returns in the short run, more variance on the state price density, time and state varying (instead of constant) risk sharing rules, as well as more important and procyclical trading volumes. These features seem consistent with the actual (or desirable) behavior of financial markets and only result from the introduction of unbiased disagreement.