Abstract: We give a comprehensive review about methods, results and open problems. Even though there has been impressive development in the last few years, all peculiar features of the Parisi spontaneous replica symmetry breaking continue to be a challenge for a complete mathematical understanding. The main core of the problem will be carefully investigated.

Abstract: Branching processes exhibit a particularly rich longtime behaviour when evolving in a random environment. Then the transition from subcriticality to supercriticality proceeds in several steps, and there occurs a second `transition' in the subcritical phase (besides the phase-transition from (sub)criticality to supercriticality). Here we present and discuss limit laws for branching processes in critical and subcritical i.i.d. environment. The results rely on a stimulating interplay between branching process theory and random walk theory.

Abstract: Recent progress on the exponential ergodicity and V-uniform ergodicity for stochastic infinite-dimensional systems is reviewed. The main contribution are results which provide exponential rate of convergence to invariant measure in the norm of total variation of merasures. Special attention is given to stochastic semilinear equations, in particular to reaction-diffusion equations, to stochastic Burgers and 2D Navier-Stokes equations. Some consequences for $L^p$-exponential ergodicity and spectral gap are given. The talk is based on joint results with Ben Goldys.

Abstract: We describe work in progress with Leonid Mytnik and Jeremy Quastel. The KPP equation is a standard model for the study of traveling waves. A large class of initial conditions yield solutions which converge to a limiting shape, which moves with constant velocity. Adding noise to the equation may give a stationary ensemble of shapes, with an average speed which is different than the speed of the deterministic wave. Brunet and Derrida made some surprising conjecturs about the speed of the wave in the random case, when the noise is small. Conlon and Doering have given an inequality which verifies half of the conjecture. We describe an approach which, if successful, will give the other half of the conjecture.

Abstract: I will give an overview of the mathematical problems that arise when imaging in a cluttered environment. I will also explain why the usual migration methods fail in this context, and why statistical methods based on constructing correlations of the data can provide good images when they are used adaptively. This means that as the data provides an image, its quality can be improved with algorithms that iteratively re-assess this quality and produce images that are incrementally better. I will show and discuss the results of several numerical simulations.

Abstract: We give a stochastic representation to the solutions to a large class of semi-linear parabolic PDEs in terms of expectations of suitable random variables over a tree of branching particles.
Since in most cases the expectations converge only for small times, we provide an integrability condition by means of a companion scalar PDE, the comparison equation. This condition in some cases characterises completely the convergence of expectations.
In order to overcome the problem of integrability, the branching tree is suitably pruned. This provides an approximated stochastic representation that in some cases converges, globally in time, to the solution of the original PDE.

Abstract: In this talk we are concerned with a second order differential operator in a Hilbert space. We prove the validity of Maximum Principle both for elliptic and parabolic equations associated to such operator. The main tools used are Preiss' theorem on differentiability of Lipschitz functions in infinite dimensions and a new version of the Asplund theorem.

Abstract: In mathematical population genetics, the forward time evolution of allele frequencies is described by S(P)DE type models like that of Fisher-Wright or Fleming-Viot, whereas the underlying genealogy of ancestral lineages, viewed backwards in time, is described by coalescent type models. In the talk we focus on a situation in which a structured coalescent in a random background comes up naturally: For a genetic locus carrying a strongly beneficial allele which has just fixed in a large population we study the ancestry at a linked neutral locus. During this ''selective sweep'' the linkage between the two loci is broken up by recombination, and the ancestry at the neutral locus is modelled by a structured coalescent in in the random background given by the advantageuos allele's frequency at the selective locus.. For large selection coefficients and under an appropriate scaling of the recombination rate, we describe and interprete an approximation to the sample partition with respect to identity by descent from the beginning of the sweep. This refines a formula going back to the pioneering work of Maynard Smith and Haigh, and complements recent work of Schweinsberg and Durrett on selective sweeps in the Moran model. This is joint work with Alison Etheridge (Oxford) and Peter Pfaffelhuber (Munich); details can be found in the preprint "An approximate sampling formula under genetic hitchhiking" (March 2005, http://arxiv.org/abs/math.PR/0503485 ).

Abstract: We study a stochastic PDE driven by space time white noise in 1 space dimension, whose invariant measure is the law of the reflecting Brownian motion. This equation is expected to be the limit of the fluctuations of a nabla phi interface model with sticky reflection at a wall.