# DESCRIPTIVE SET THEORY IN PARIS

## Titles and Abstracts of the talks

Aurélien ALVAREZ                Orbit equivalence and free product decompositions

Spiros ARGYROS                    Connecting Heterogeneous Structures in Banach Space Theory

Riccardo CAMERLO                Quotients of projective Fraïssé limits

Márton ELEKES                        Solvability cardinals

Petr HOLICKY                          Uniformizations of sets with large sections

Vassilis KANELLOPOULOS    Hausdorff measures associated to functions of square bounded variation.

Jordi LOPEZ ABAD                  Some Examples of Generic Banach Spaces

Witold MARCISZEWSKI          On Banach spaces whose norm-open sets are weakly $F_\sigma$-sets

Julien MELLERAY                    How to recognize an isometry?

Luca MOTTO ROS                    Analytic equivalence relations and bi-embeddability.

Christian ROSENDAL                A simple proof of Gowers' dichotomy theorem

Brian SEMMES                           A game for the Borel functions

Todor TSANKOV                       Subequivalence relations and positive definite functions

Miroslav ZELENY                       Additive families of low Borel classes

Aurélien ALVAREZ                Orbit equivalence and free product decompositions

If a countable group acts freely, measure preservingly and ergodically on the standard non-atomic probability space, it gives rise to a type II_1 measure equivalence relation. There is a well defined notion of free product decomposition for such an equivalence relation and we are interested in rigidity results in that context. Namely if two groups admiting free product decompositions give rise to the same measure equivalence relation, does there exist a one-to-one correspondance between the measure equivalence relations induced by the factors ? We shall discuss and give results to this question.

Riccardo CAMERLO                Quotients of projective Fraïssé limits

In this talk I will characterise all topological spaces that can be obtained as quotients of projective Fraïssé limits of a projective Fraïssé family of finite structures with a binary relation.

Márton ELEKES                        Solvability cardinals

The following research was originally motivated by the theory of liftings. This is a joint work with M. Laczkovich.

Let $\mathbb{R}^\mathbb{R}$ denote the set of real valued functions defined on the real line.
A map $D: \mathbb{R}^\mathbb{R} \to \mathbb{R}^\mathbb{R}$ is said to be a {\it difference operator}, if there are real numbers $a_i , b_i \ (i=1,\ldots , n)$ such that $(Df)(x)=\sum_{i=1}^n a_i f(x+b_i)$ for every $f\in \mathbb{R}^\mathbb{R}$ and $x\in \mathbb{R}$.
By a {\it system of difference equations} we mean a set of equations
$S=\{ D_i f=g_i : i\in I\}$, where $I$ is an arbitrary set of indices, $D_i$ is a difference operator and $g_i$ is a given function for
every $i\in I$, and $f$ is the unknown function.
One can prove that a system $S$ is solvable if and only if every finite subsystem of $S$ is solvable.
However, if we look for solutions belonging to a given class of functions then the analogous statement is no longer true. For example, there exists a system $S$ such that every finite subsystem of $S$ has a solution which is a trigonometric polynomial, but $S$ has no such solutions; in fact, $S$ has no measurable solutions.

This phenomenon motivates the following definition. Let ${\cal F}$ be a class of functions. The {\it solvability cardinal} ${\rm sc} (\cal F)$ of ${\cal F}$ is the smallest cardinal number $\kappa$ such that whenever $S$ is a system of difference equations and each subsystem of $S$ of cardinality
ess than $\kappa$ has a solution in ${\cal F}$, then $S$ itself has a solution in ${\cal F}$.

In this talk we will present the solvability cardinals of most function classes that occur in analysis. As it turns out, the behaviour of ${\rm sc} (\cal F)$ is rather erratic. For example, ${\rm sc} (\textrm{polynomials})=3$ but ${\rm sc} (\textrm{trigonometric polynomials})=\omega _1$,
${\rm sc} (\{ f: f\ \textrm{is continuous}\}) = \omega_1$ but ${\rm sc} (\{ f: f\ \textrm{is Darboux}\}) =(2^\omega )^+$, and ${\rm sc} (\mathbb{R}^\mathbb{R} )=\omega$.

We will also consistently determine solvability cardinals of the classes of Borel, Lebesgue and Baire measurable
functions, and give some partial answers for the Baire class 1 and Baire class $\alpha$ functions. We will also pose the intriguing open problem concerning the possibility of defining natural ranks for the Baire class $\alpha$ functions.

Vassilis KANELLOPOULOS    Hausdorff measures associated to functions of square bounded variation.

For each function of square bounded variation we define a Hausdorff-type measure on the unit interval. We will present some properties and applications of this measure to the structure of subspaces of James Function space.

Witold MARCISZEWSKI          On Banach spaces whose norm-open sets are weakly $F_\sigma$-sets

We will discuss the results concerning non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is $\sigma$-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces $C(K)$ with a $\sigma$-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina.
concerning Borel structures and Kadec renormings in Banach spaces.

Julien MELLERAY                    How to recognize an isometry?

I'll discuss what's known about the following problem: given a homeomorphism f of a Polish metric space X, when is there a compatible distance d on X such that f is a d-isometry?
Note: the question mark in the title is important - I definitely don't have an answer to that question!

Luca MOTTO ROS                    Analytic equivalence relations and bi-embeddability.

The analysis of the structure of the analytic equivalence relations (ER for short) under Borel-reducibility has been one of the most relevant subject in the recent history of Descriptive Set Theory. Among ERs, a special place is occupied by the isomorphism relation on countable models of a certain $\mathcal{L}_{\omega_1 \omega}$-sentence $\varphi$. However, isomorphism relations are just a special case of ER, as there are many example of ERs which are not Borel-reducible to an isomorphism relation. On the contrary, we will show in this talk that the bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) relation is able to capture the whole complexity of the ER structure: for every ER $E$ there is an $\mathcal{L}_{\omega_1 \omega}$-sentence $\varphi$ such that $E$ is Borel-equivalent to bi-embeddability (resp. bi-homomorphism, bi-weak-homomorphism) on the collection of countable models of $\varphi$. This is joint work with Sy D. Friedman.

Christian ROSENDAL                A simple proof of Gowers' dichotomy theorem

We prove an exact, i.e., formulated without Delta-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers’ block sequence game and winning strategies in the finite asymptotic game. This allows us to recover Gowers’ dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.

Brian SEMMES                           A game for the Borel functions

In this talk, I summarize my thesis work.
I introduce a Wadge-style game G(f) which characterizes the Borel functions on the Baire space, in the sense that Player II has a winning strategy in G(f) iff f is Borel.  By tinkering with the rules of this game, it is possible to characterize subclasses of Borel functions, including Baire classes 1 and 2.  Using game-theoretic methods, I prove decomposition theorems for two subclasses of Baire class 2, extending a result of Jayne and Rogers (1982).

Miroslav ZELENY                       Additive families of low Borel classes

An important conjecture in the theory of Borel sets in non-separable metric spaces is whether any point-countable Borel-additive family in a complete metric spaces has a $\sigma$-discrete refinement. We confirm the conjecture for point-countable $F_{\sigma\delta}$-additive families, generalizing thus the result of R.W. Hansell.