HColl_2016-2017

Take a group, your favorite one, for example the group of $n$ by $n$ matrices with integer coefficients and determinant $1$ . Take your favorite space, for instance the real vector space of dimension $3$ , or the unit circle, or a complex projective variety. Does your favorite group act (faithfully) on your favorite space ? Does it act by homeomorphisms, or by diffeomorphisms, or by algebraic transformations? I shall address this type of questions, focussing on simple examples.

Isabelle Gallagher (Université Paris - Diderot): From particle systems to Fluid Mechanics

The question of deriving Fluid Mechanics equations from deterministic systems of interacting particles obeying Newton's laws, in the limit when the number of particles goes to infinity, is a longstanding open question suggested by Hilbert in his 6th problem. One of the challenges behind this program is understanding the appearance of irreversibility in the limiting process. In this talk we shall present a few attempts in this program, by explaining how to derive some linear models such as the Stokes-Fourier equations. This corresponds to joint works with Thierry Bodineau and Laure Saint Raymond.

Stefano Olla (Université Paris - Dauphine): Macroscopic Heat Transport from Microscopic Dynamics

Macroscopic heat flux in dielectric materials is related to the transfer of energy distributed on high frequencies (thermal energy) and is governed by heat equation or other diffusive or superdiffusive equations, depending on the microscopic structure of the material. I will review some mathematical models of microscopic dynamics where evolution equations for the temperature profile emerge through space-time scaling limits.

Petra Schwer (Karlsruher Institut für Technologie): Combinatorics of Coxeter groups and Affine Deligne-Lusztig varieties

Coxeter groups are abstract reflection groups, each of which has a geometric presentation as a reflection group of a sphere, Euclidean space or hyperbolic space.

In this talk we will highlight the geometric and combinatorial behaviour of spherical and affine Coxeter groups and their associated Coxeter complexes. Folded galleries and root operators, as appearing in the Littelmann path model, will be in the focus. Together with Liz Milicevic and Anne Thomas we developed a new approach to study dimensions of affine Deligne-Lusztig varieties (ADLVs), using these combinatorial methods in the Coxeter complexes. We will explain the main ideas of this approach and present the current state of the art concerning nonemptiness and dimensions of ADLVs.

Hendrik Weber (University of Warwick): Recent progress in singular stochastic PDE

This talk is concerned with stochastic partial differential equations (SPDE) driven by a singular noise term, such as space-time white noise. Following Hairer’s groundbreaking work on regularity structures research on these equations has witnessed an enormous activity over the last years and the aim of the talk is to give a survey of some of the recent results.

First I will explain how non-linear SPDE arise as scaling limits of discrete models from statistical mechanics. Then I will outline how the phenomenon of metastability can be observed in such an equation. Finally, if time permits, I will try to convey an idea of the type of argument which allows to analyse these SPDE.