Invited lectures on 47th Winter School
Gilles Godefroy, Institut de Mathématiques 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 Grothendieck’s 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, Université 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