Past

In recent years, the use of random planar maps as discretized random surfaces has received a considerable attention in the physicists community. It is believed that the large-scale properties, or the scaling limit of these objects should not depend on the local properties of these maps, a phenomenon called universality.

By using a bijection due to Bouttier-di Francesco-Guitter between certain classes of planar maps and certain decorated trees, we give instances of such universality

phenomenons when the random maps follow a Boltzmann distribution where each face with degree $2i$ receives a nonnegative weight $q(i)$. For example, we show that under

certain regularity hypothesis for the weight sequence, the radius of the random map conditioned to have $n$ faces scales as $n^{1/4}$, as predicted by physicists and shown in the case of quadrangulations by Chassaing and Schaeffer. Our main tool is a new invariance principle for multitype Galton-Watson trees and discrete snakes.

The aging of spin-glasses has been of much interest in the last decades. Since its explanation in the context of real spin-glass models is out of reach, several effective models were proposed in physics literature. In my talk I will present how aging can be rigorously proved in so called trap models and what is the mechanism leading to it. In particular I will concentrate on conditions leading to the fact that one of usual observables used in trap models converges to arc-sine law for Levy processes.

Random Walks in Dirichlet Environment play a special role among random walks in random environments since the annealed law corresponds to the law of an edge oriented reinforced random walks. We will give few results concerning the ballistic behaviour of these walks and some properties of the asymptotic velocity. We will also compare the behaviour of these walks with general random walks in random environments in the limit of small disorder

In this talk, we present several numerical issues, that we currently pursue,

related to accurate approximation of option prices. Next to the numerical

solution of the Black-Scholes equation by means of accurate finite differences

and an analytic coordinate transformation, we present results for options under

the Variance Gamma Process with a grid transformation. The techniques are

evaluated for European and American options.

The famous Eckart-Young Theorem states that the (normwise) condition number of a matrix is equal to the reciprocal of its distance to the nearest singular matrix. In a recent paper we proved an extension of this to a number of structures common in matrix analysis, i.e. the structured condition number is equal to the reciprocal of the structured distance to the nearest singular matrix. In this talk we present a number of related results on structured eigenvalue perturbations and structured pseudospectra, for normwise and for componentwise perturbations.

We show that the solutions of stochastic differential equations converge in

the rough path metric as the coefficients of these equations converge in a

suitable lipschitz norm. We then use this fact to obtain results about

differential equations driven by the Brownian rough path.

It is now known that the overall behaviour of a simple random walk (SRW) on

supercritical (p>p_c) percolation cluster in Z^d is similiar to that of the SRW

in Z^d. The critical case (p=p_c) is much harder, and one needs to define the

'incipient infinite cluster' (IIC). Alexander and Orbach conjectured in 1982

that the return probability for the SRW on the IIC after n steps decays like

n^{2/3} in any dimension. The easiest case is that of trees; this was studied by

Kesten in 1986, but we can now revisit this problem with new techniques.

We use the partial differential inclusion method to establish existence of

infinitely many weak solutions to the one-dimensional version of the

Perona-Malek anisotropic diffusion model in the theory of image processing. We

consider the homogeneous Neumann problem as the model requires.

.

The Yang-Mills energy is a non-negative functional on the space of connections on a principal bundle over a Riemannian manifold. At a heuristical level, this energy determines a Gibbs measure which is called the Yang-Mills measure. When the manifold is a surface, a stochastic process can be constructed - at least in two different ways - which is a sensible candidate for the random holonomy of a connection distributed according to the Yang-Mills measure. This process is constructed by using some specifications given by physicists of its distribution, namely some of its finite-dimensional marginals, which of course physicists have derived from the Yang-Mills energy, but by non-rigorous arguments. Without assuming any familiarity with this stochastic process, I will present a large deviations result which is the first rigorous link between the Yang-Mills energy and the Yang-Mills measure.