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Autumn School on Nonlinear Geometry of Banach Spaces and Applications

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Dear colleague,

we would like to announce the upcoming Autumn School on Nonlinear Geometry of Banach Spaces and Applications, to be held in Métabief from the 19th to the 24th of October 2014.
The following mathematicians have kindly accepted our invitation to deliver a short course (move the mouse pointer over the title to see the abstract):

Gilles Godefroy, Université Paris 6
Lipschitz free spaces
The Lipschitz free space associated with a metric space $M$ is the predual of the space of real valued Lipschitz functions on $M$. The simplicity of this definition hides the complexity of this family of Banach spaces, which is still not well understood. We will present the basic results together with situations where these spaces are actually used. Then we will discuss some of the many open questions in this theory. This line of research is quite recent and it is possible to attack it with relatively few technical tools. Thus, these lectures should give a rather elementary approach to the open problems.
Petr Hájek, Czech Academy of Sciences and Czech Technical University
Polynomials on Banach spaces
We will introduce the abstract concept of polynomial on a Banach space. We will study the behaviour of polynomials in connection with the linear structure of the underlying Banach space. Among the main results we will indicate the proof of Deville\’s theorem on the containment of $\ell_p$ subspaces for Banach spaces with a separating polynomial, and the theory of polynomial algebras developped by S. D’Alessandro, M. Johanis and the present author.
PDF - 494 kb
Lecture notes



Mikhail Ostrovskii, St.-John’s University
Metric characterizations of some classes of Banach spaces
Study of metric characterizations of different classes of Banach spaces is inspired by the following important result of Ribe (1976): If two Banach spaces are uniformly homeomorphic, then they are crudely finitely representable in each other. This result implies that classes of Banach spaces which are both isomorphic invariant and are local, that is, determined by finite-dimensional subspaces, are preserved under uniform homeomorphisms and hence it is plausible that they can be characterized in purely metric terms. If such characterization is found, one can try to transfer some of the theory known for the corresponding class of Banach spaces to metric spaces. This direction of research was suggested by Lindenstrauss (unpublished) and Bourgain (1986), and became known as Ribe program. This direction of research was presented in a short survey of Ball (2013) and an extensive survey of Naor (2012). In these talks, after a summary of the known results on metric characterizations of well-known local isomorphic invariants and a short list of open problems, I plan to present more recent results on metric characterization of superreflexivity.

Metric characterizations can be studied for isomorphic invariants which are not local, such as reflexivity and the Radon-Nikodým property (RNP). In these talks, after a short summary of known results on metric characterization of non-local properties, I plan to focus on metric characterization of the RNP.

The RNP is one of the most basic and important isomorphic invariants of Banach spaces. A Banach space $X$ is said to have the Radon-Nikodým property if an analogue of the Radon-Nikodým theorem holds for $X$-valued measures which are absolutely continuous with respect to some real-valued measures. This property can be characterized in many different analytic, geometric, and probabilistic ways. Possibly the most well-known is the characterization in terms of Lipschitz functions (goes back to Clarkson (1936) and Gelfand (1938)): A Banach space $X$ has the RNP if and only if each $X$-valued Lipschitz function on the real line is almost everywhere differentiable. The RNP plays an important role in the theory of metric embeddings (works of Cheeger, Kleiner, Lee, and Naor (2006--2009)). In this connection Johnson (2009) suggested the problem of metric characterization of RNP. I am going to present a solution of this problem in terms of thick families of geodesics.
Nirina Lovasoa Randrianarivony, Saint Louis University
Nonlinear quotients and asymptotic geometry of Banach spaces
A uniform quotient map is a map $f:X \to Y$ between metric spaces that is uniformly continuous and uniformly open in the sense that there exists a nondecreasing map $\omega: (0,\infty) \to (0,\infty)$ such that for every $x \in X$ and every $r>0$, $ B_Y ( f(x),\omega (r)) \subseteq f \left( B_X (x,r) \right)$. If $f$ is surjective, then $Y$ is called a uniform quotient of $X$.

A technique used for the study of uniform quotient maps leads naturally to the study of an asymptotic geometry of Banach spaces called property $(\beta)$. We will present this technique, as well as give a survey of property $(\beta)$ as found in the literature. We will then present a result on the preservation of $(\beta)$-renormability under uniform quotient maps between separable Banach spaces.
Alain Valette, Université Neuchâtel
Embeddings of groups into Hilbert spaces
We will cover the following topics:
  1. Groups as metric spaces (word length, Cayley graphs, amenability)
  2. Coarse embeddings
  3. The Novikov conjecture
  4. Yu's property (A)

The purpose of this meeting is to bring together researchers and students with common interest in the field. The meeting will offer many possibilities for informal discussions.
Graduate students and others beginning their mathematical career are encouraged to participate.
This school is followed by Conference on Geometric Functional Analysis and its Applications in Besançon and is part of the special trimester "Geometric and non-commutative methods in functional analysis" which will be organized at Université Franche-Comté.

The registration for both events is now closed. The deadline for the registration is September 5, 2014. Notice though that the capacity of the venue might be reached before the deadline, in which case people who registered earlier will be given priority.

Kindly inform your colleagues and students who you think might be interested.

We are looking forward to meeting you in Franche-Comté

Gilles Lancien, Tony Procházka
gilles.lancien(at)univ-fcomte.fr antonin.prochazka(at)univ-fcomte.fr