HColl_2019

In this lecture I shall discuss how kinetic theory has found applications in various subjects such as the mathematical theory of neurosciences. It is a surprising encounter of three fields of research, namely multiscale dynamical systems, statistical physics and mathematical neurosciences. During this lecture, it will be the opportunity to review some recent developments in mean field theory and relative entropy techniques for kinetic models. The talk is intended to be accessible to a generalist scientific audience.

Jesper Grodal (University of Copenhagen): Local to global in modular representation theory and homotopy theory

Many questions in mathematics evolve around passing from "local" information to "global" information. In modular representation theory, long-standing conjectures predict how representations of a group relate to representations of "local" subgroups, though the exact nature of the proposed bijection is often mysterious. One of the strengths of homotopy theory is that it can allow for more creative sorts of induction, or gluing, taking into account also higher order structure. My talk will describe one success of this viewpoint, in the classification of so-called endotrivial modules, which can be though of as "almost-1-dimensional" modules. I will tell this story from the beginning, starting with work of Dade in the 70s, and leading into the present...

Bénédicte Haas (Université Paris XIII): Markov-branching trees and their scaling limits

We will survey some of the recent progress on the description of large-scale structure of random trees. Describing the structure of large (random) trees, and more generally large graphs, is an important goal of modern probabilities and combinatorics. We will focus on sequences of random trees that satisfy a certain Markov–Branching property, which appears naturally in a large set of models. This property is a sort of discrete fragmentation property which roughly says that in each tree of the sequence, the subtrees above a given height are independent with a law that depends only on their total size. Under appropriate assumptions, we will describe the scaling limits of such sequences of trees, and then discuss several applications.

Federico Rrodriguez Hertz (Pennsylvania State University): Rigidity in hyperbolic systems.

In the 80's two type of related rigidity results emerged, one of geometric flavor and the other dynamical, but both have a dynamical background. Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. And de la Llave, Marco and Moriyon showed that for two dimensional Anosov diffeomorphisms, the marked Lyapunov spectrum determines the smooth isomorphism type of the system. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theory can be developed. This project is joint with A. Gogolev.

Stefan Luckhaus (Universität Leipzig): Geometric aspects of the plastic deformation of metals

There is an intimate connection between Riemannian Geometry and nonlinear Elasticity. Elastic strain can be viewed as a flat connection in an affine bundle.As applied mathematicians we are interested in the homogenisation of geometric defects and ultimately the space time structure on many scales atomistic ,sub grain,grain and macroscopic . The results I can present so far link geometric rigidity as developed by Friesecke, James, Mueller to grain boundary theories by Reid and Schockley

Victor Nistor (Université de Lorraine): Lie groups and spectral theory on singular and noncompact spaces

Many questions in geometry, number theory, and other areas of mathematics and its applications involve non-compact manifolds with nice ends. An approach that applies to a large class of such manifolds is via the Lie theory of vector fields on a certain compactification to a manifold with corners. This leads us to the class of "Lie manifolds," which includes the classes of asymptotically hyperbolic and asymptotically euclidean manifolds. A typical question that one can answer is that of determining the essential spectrum of suitable geometric operators. This recovers, in particular, the classical HVZ theorem on the essential spectrum of the N-body hamiltonian in Quantum Mechanics.

One of the simplest, but most instructive examples of manifolds with nice ends (Lie manifolds) is that of the class of manifolds with cylindrical ends, which arises in the Atiyah-Patodi-Singer index theorem and in the related results of Kondratiev on manifolds with conical points. I will therefore begin by discussing these results. Then I will report on the Lie algebra approach to non-compact manifolds, as well as on some more recent results on the related question of determining the index of Fredholm operators on Lie manifolds due to K. Bohlen and collaborators. I will also quickly mention some connections with bounded geometry and to singular spaces (which are treated by reducing to the non-compact case). Most of the results presented are based on joint works with Amnann, Grosse, Monthubert, and Prudhon.