# THEMATIC PROGRAMS

October 25, 2016

## Thematic Program on Set Theory and Analysis

### Workshop on Geometry of Banach spaces and infinite dimensional Ramsey theory November 11 - 15, 2002

 Dale Alspach Dominique Lecomte George Androulakis Denny Leung Spiros Argyros Jordi Lopez-Abad Alexander D. Arvanitakis Edward Odell Valentin Ferenczi Anna Pelczar Gilles Godefroy Haskell Rosenthal Petr Hajek Bunyamin Sari Jose Iovino Thomas Schlumprecht Vassilis Kanellopoulos Robert Kaufman

Dale Alspach (Oklahoma State) Bibliography
Some Ordinal Indices in Banach Space Theory

Ordinal indices have been used in several ways to quantify behaviors and structures in infinite dimensional Banach spaces. In this talk I will survey some results which make use of ordinal indices. Among those indices discussed will be the Szlenk index, Bourgain's $\ell_1$ index, and the $\ell_1^+$ weakly null index. Along the way I will define the Schreier sets and indicate the role that they have played in the theory. While the talk will be far from exhaustive, I hope that the selection will give non-experts in Banach space theory a useful introduction.

Valentin Ferenczi, (Paris 6)
On a question by Haskell P. Rosenthal

This is a work in collaboration with A. Pelczar and C. Rosendal. We give partial answers to the following question by Haskell P. Rosenthal: If a basis $(e_n)$ of a Banach space is such that every block basis of $(e_n)$ has a subsequence equivalent to $(e_n)$, must $(e_n)$ be equivalent to the unit vector basis of $c_0$ or $l_p$, $p \in [1,+\infty)$?

Gilles Godefroy, (Pierre et Marie Curie)
Rosenthal compact sets and analytic filters

We present in this talk a rather old result on characterizations of Rosenthal compact sets, in terms of analyticity of the filter of neighbourhoods of the diagonal in the square space. Several applications and connections to more recent results will be mentionned.

Petr Hajek, (Czech Academy of Science)
Quantitative Krein theorem

A quantitative version of a classical theorem of Krein, stating that closed convex hull of a weak compact is also a weak compact will be given.

Robert Kaufman, (Illinois at Urbana-Champaign)
Complexity of a certain set of norms in a Banach space.

Let X be a separable, infinite-dimensional Banach space and
N(X) the set of equivalent norms in X. An element x of X is said to be a "co-smooth" direction of a morm, if that norm is Gateaux-smooth everywhere in the direction of x. Let G(0) be the set of norms admitting at least one co-smooth direction x (not 0). What is the complexity of G(0)? Counting quantifiers leads to the class PCA, but this is too large. The set G(0) can be obtained by applying the Hausdorff operation A to a scheme of co-analytic sets. Moreover, even when X is not too exotic, the class G(0) reduces every set of the specified class by a continuous map into N(X).

Dominique Lecomte, (Pierre et Marie Curie)
How can we recover Baire class one functions ?

Let X and Y be separable metrizable spaces, and f:X --> Y be a map. We want to recover f from its values on a small set via a simple algorithm. We give sufficient conditions on X to insure this when f is Baire class one. We study the limits of these results. This leads us to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

Edward Odell, (Texas at Austin)
Ramsey Methods in Banach spaces

This is an expository talk concerning the use of Ramsey theory in the study of infinite dimensional Banach spaces. We will discuss a number of the results mentioned in the workshop Rationale as well an some open problems.

Anna Pelczar, (Jagiellonian University)
Combinatorial techniques in the classification of Banach spaces

We will present results along the lines the classification of Banach spaces proposed by T.Gowers - in terms of richness of the space of operators on a given Banach space. The method used in proofs extends the technic of B.Maurey's proof of Gowers dichotomy for hereditarily indecomposable spaces and unconditional sequences.

Bunyamin Sari (Alberta)
On spreading models of Orlicz sequence spaces

We consider the partially ordered set which consists of spreading models an Orlicz sequence space together with the partial order defined with respect to domination of bases. A description of this partially ordered set along with some illustrative examples will be presented.

Thomas Schlumprecht, (Texas A&M)
Asymptotic Structure and Games of higher complexity in a Banach space.

We introduce for a countable ordinal $\alpha$ the the asymptotic structure $\{X\}_\alpha$, similar as it was done in the finite case by Maurey, Milmann, and Tomczak-Jaegermann. We are doing this using finite games of complexity $\alpha$. With these tools we deduce certain quantitative versions of Gowers' dichotomy result. We also give a different proof of the combinatorical theorem underlying this dichotomy result.