Past

In this talk we will discuss a theory of divergence-measure fields and related

geometric measures, developed recently, and its applications to some fundamental

issues in mathematical continuum physics and nonlinear conservation laws whose

solutions have very weak regularity, including hyperbolic conservation laws,

degenerate parabolic equations, degenerate elliptic equations, among others.

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence.

Moving hydrodynamic boundaries (waves and bubbles, for example) produce acoustic signatures.

It is well known that the computation of a few extreme eigenvalues, and the corresponding eigenvectors, of a symmetric matrix A can be rewritten as computing extrema of the Rayleigh quotient of A. However, since the Rayleigh quotient is a homogeneous function of degree zero, its extremizers are not isolated. This difficulty can be remedied by restricting the search space to a well-chosen manifold, which brings the extreme eigenvalue problem into the realm of optimization on manifolds. In this presentation, I will show how a recently-proposed generalization of trust-region methods to Riemannian manifolds applies to this problem, and how the resulting algorithms compare with existing ones.

I will also show how the Joint Diagonalization problem (that is, approximately diagonalizing a collection of symmetric matrices via a congruence transformation) can be tackled by a differential geometric approach. This problem has an important application in Independent Component Analysis.

For atomistic (and related) material models, global minimization

gives the wrong qualitative behaviour; a theory of equilibrium

solutions needs to be defined in different terms. In this talk, a

process based on gradient flow evolutions is presented, to describe

local minimization for simple atomistic models based on the Lennard-

Jones potential. As an application, it is shown that an atomistic

gradient flow evolution converges to a gradient flow of a continuum

energy, as the spacing between the atoms tends to zero. In addition,

the convergence of the resulting equilibria is investigated, in the

case of both elastic deformation and fracture.

Levy trees are random continuous trees that are obtained as

scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in

PTRF) math.PR/0509518 (preprint)

math.PR/0509690 (preprint).

I will present two-sided estimates for the heat kernel of the elliptic

Schr