# Hausdorff-Kolloquium 2011/2012

**Date:** November 02, 2011 - January 25, 2012

**Venue:** Mathematik-Zentrum, Lipschitz Lecture Hall, Endenicher Allee 60, Bonn

## Wednesday, November 2

**15:15 **Curtis T. McMullen (Harvard University, currently MPIM Bonn): Billiards and moduli spaces

16:45 Camillo de Lellis (Universität Zürich): h-principle and fluid dynamics

## Wednesday, November 30

**15:15 **Ben Green (University of Cambridge): Approximate groups

16:45 Frank Merle (Cergy-Pointoise and IHES): Recent developments on the global behavior to critical nonlinear wave equation

## Wednesday, December 21

**15:15 **Alessio Figalli (University of Texas, Austin): On the Ma-Trudinger-Wang condition

16:45 Francois Labourie (Universite Paris-Sud, Orsay): Swapping points on the circle

## Wednesday, January 25

**15:15 **Nicolas Bergeron (Universite Pierre et Marie Curie, Jussieu): On the growth of Betti numbers of locally symmetric spaces

16:45 Stefan Grimme (Universität Bonn): Accurate density functionals for the chemesty of large molecules

## Abstracts:

Alessio Figalli: On the Ma-Trudinger-Wang condition

The Ma-Trudinger-Wang condition, first introduced to prove regularity of optimal transport maps for general cost functions, has turned out to be a useful tool for:

- Obtaining geometric informations on the underlying manifold.

- Making the principal-agent problem theoretically and computationally tractable, allowing to derive uniqueness and stability of the principal's optimum strategy. In this expository talk I'll give an overview of these results.

Ben Green: Approximate groups

A finite set A in a group G is a subgroup if and only if xy^{-1} lies in A whenever x and y do. But what if xy^{-1} only lies in A some of the time? This is one possible notion of what it means for A to be an approximate group. The aim of my talk is to introduce this notion, give some examples, explain what we know and do not know about approximate groups, and then discuss some applications (expanders, growth in groups and, if I dare, differential geometry)

Francois Labourie: Swapping points on the circle

We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra -- called the algebra of multifractions -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on certain character varieties as well as on the space of SL(n;R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd-Sokolov structure and to the Atiyah-Bott-Goldman symplectic structure for classical Teichmüller spaces and Hitchin components.

Camillo de Lellis: h-principle and fluid dynamics

There are nontrivial solutions of the incompressible Euler equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. There are C^1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the threedimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. I will discuss the corresponding mathematical theorems, point out some surprising relations and give evidences that, maybe, they are not merely a mathematical game.