Winter School

Invited lectures on 47th Winter School

Gilles Godefroy, Institut de Mathmatiques de Jussieu - Paris Rive Gauche, Paris, France

     Series of 3 lectures: Lipschitz isomorphisms and uniform isomorphisms between Banach spaces

Non-linear geometry of Banach spaces is a relatively new but rapidly expanding field of research. One of its goals is to determine how much of the linear structure of a Banach spaces can be recovered from a knowledge of its metric or its uniform structure. In other words, if two Banach spaces are Lipschitz-isomorphic, or merely uniformly isomorphic, are they linearly isomorphic? Or at least, do they share some significant isomorphic properties? We will devote three lectures to this field, which will be organized as follows.
1. The free space of a metric space.
A simple and canonical construction permits to associate to every metric space $M$ a Banach space $\mathcal{F}(M)$ in such a way that Lipschitz maps between metric spaces become linear maps between their corresponding free spaces. We will display this construction and see how to use the free spaces. In particular, they will provide canonical examples of Lipschitz isomorphic (non separable!) Banach spaces which are not linearly isomorphic.
2. The Gorelik principle.
The Gorelik principle is a topological result which roughly asserts that a Lipschitz isomorphism from $X$ to $Y$ maps a large enough ball of any subspace of finite codimension of $X$ to a subset of $Y$ of compact codimension. When properly used, this principle provides a substitute to the fact that Lipschitz isomorphisms cannot be transposed to isomorphisms between the dual spaces. We will see how to use it for showing that a space which is Lipschitz-isomorphic to $c_0$ is actually linearly isomorphic to $c_0$.
3. Approximation properties.
A proper use of free spaces show that Grothendiecks bounded approximation property (BAP) is stable under Lipschitz-Isomorphisms. We will further study the interplay between BAP and extension of Lispchitz maps. For instance, a canonical approach permits to relate the existence of a Banach spaces failing BAP and the existence of Banach-space valued Lipschitz functions on a subset of a metric space $M$ which do not admit Lipschitz extensions to $M$.
In all these lectures, related open questions will be recalled and commented. An effort will be made to avoid technicalities.

Todor Tsankov, Universit Paris, France

     Series of 3 lectures: Analysis on automorphism groups of countable structures

The classical theory of harmonic analysis and dynamical systems is usually restricted to actions of locally compact groups but in recent years, some of the theory has been extended to more general Polish groups. In this series of talks, I will concentrate on closed subgroups of S_infty, which is perhaps the best understood class of non-locally compact groups in this context, and in the study of which model-theoretic tools play an important role. I will explain some aspects of the theory concerning the unitary representations of some of those groups and their measure-preserving actions.

Anush Tserunyan, University of Illinois at Urbana-Champaign, USA

     Series of 2-3 lectures: Countable Borel equivalence relations

An equivalence relation $E$ on a Polish space $X$ (e.g., $X := \mathbb{R}$, $L^1(\mathbb{R})$, $\mathbb{N}^\mathbb{N}$) is \emph{Borel} if it is a Borel subset of $X^2$, and it is \emph{countable} if each $E$-class is countable. Countable Borel equivalence relations (CBERs) naturally arise as orbit equivalence relations of Borel actions of countable groups (e.g., $\mathbb{Q} \curvearrowright \mathbb{R}$ by translation). From another, rather combinatorial, angle, a CBER $E$ can always be viewed as the connectedness relation of a locally countable Borel graph (e.g., take $E$ as the set of edges). These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability, descriptive graph combinatorics, and geometric group theory.
These lectures will feature this interplay. On one hand, we will learn some tools from each of the aforementioned subjects to analyze the structure of CBERs. On the other hand, we will utilize the basic theory of CBERs to prove some well-known results in those subjects