# DESCRIPTIVE SET THEORY IN PARIS

## Titles and Abstracts of the talks

 Luca MOTTO ROS Isomorphism and bi-embeddability Julien MELLERAY Applications of continuous logic to the study of topological groups. Christian ROSENDAL Polish groups acting on trees Jordi LOPEZ ABAD Examples of non-separable L1(µ)-preduals. Damien GABORIAU Measured group theory and orbit equivalence Su GAO Variations of 2-colorings on countable groups Pandelis DODOS Fixing properties of operators and applications. Simon THOMAS Universal Borel Actions of Countable Groups Asger TÖRNQUIST Turbulence and the classification of ITPFI factors. Lionel NGUYEN VAN THE Precompact groups and isometric actions on Banach spaces. Witold MARCISZEWSKI On some problems concerning Borel structures in function spaces Stevo TODORCEVIC Tukey Function on the Nowhere Dense Ideal

Luca MOTTO ROS               Isomorphism and bi-embeddability

Given a pair of analytic equivalence relations E and F, we determine under which conditions it is possible to find an L_{\omega_1 \omega}-elementary class C such that E and F are Borel equivalent to, respectively, isomorphism and bi-embeddability on C: quite surprisingly, it turns out that, apart from the obvious limitations, such an elementary class always exists. This result gives an (almost complete) answer to previous questions of Louveau-Rosendal (2005) and Friedman-Motto Ros (2009).

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Julien MELLERAY               Applications of continuous logic to the study of topological groups.

In this talk I will try to explain how one can use continuous logic to study large topological groups; examples discussed will include a new automatic continuity theorem, a variant of Bergman's property, and a discussion of oscillation stability in this context.

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Christian ROSENDAL                Polish groups acting on trees

Un nombre de resultats de R.M. Dudley, H. Bass, R. Alperin, S. Shelah, D. Macpherson et S. Thomas (parmi autres) traitent des divers problemes concernant des groupes polonais agaissant sur des arbres. Je vais presenter des nouveaux resultats dans cette direction qui partiellement confirment une conjecture qui aurait comme consequence une bonne partie des resultats precedents.

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Jordi LOPEZ ABAD               Examples of non-separable L1(µ)-preduals.

We present a unified method for constructing several uncontable direct limits of copies of $\ell_\infty^n$ spaces. In particular, examples of non-separable Gurarij spaces, preduals of $\ell_1(\omega_1)$ and  the dual constructions of generic simplices will be exposed. We will discuss  some striking differences between these non-separable constructions and the corresponding separable examples where the generic objects tend to be unique.

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Su GAO               Variations of 2-colorings on countable groups.

I will talk about recent joint work with Steve Jackson and Brandon Seward on various notions of 2-colorings on countable groups. A 2-coloring on a countable group is an element of the Bernoulli flow that generates a free subflow. We consider several combinatorial variations of the notion of 2-coloring and give their topological and algebraic characterizations.

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Simon THOMAS                Universal Borel Actions of Countable Groups.

If the countable group G has a nonabelian free subgroup, then there exists a standard Borel G-space such that the corresponding orbit equivalence relation is countable universal. In this talk, I will consider the question of whether the converse also holds.

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Asger TÖRNQUIST                Turbulence and the classification of ITPFI factors.

In this talk I will present a recent result (joint with Roman Sasyk, Buenos Aires) regarding the classification of those von Neumann factors that arise as the Infinite Tensor Product of Factors of type I (ITPFI). This class of factors, which has a particularly simple definition, was the first in which an uncountable family of non-isomorphic factors was found, by Powers in 1967. Later (1973) Woods showed that the classification of these factors is not smooth by, in modern terminology, Borel reducing E_0 to the isomorphism relation.
I will show that ITPFI factors are not classifiable by countable structures, using a relatively simple Baire category argument. In the process we also obtain a new and substantially less technical proof of Woods' Theorem.

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Lionel NGUYEN VAN THE                Precompact groups and isometric actions on Banach spaces.

Le but de cet exposé est de présenter une caractérisation des groupes topologiques séparables G possédant la propriété notée (FB) : Toute action isométrique affine de G sur tout espace de Banach admet un point fixe.
On montre par exemple que (FB) admet une reformulation purement topologique, et qu'il existe un espace de Banach X (construit à partir de l'espace d'Urysohn) tel que pour tout groupe topologique séparable G, G possède (FB) ssi toute action isométrique affine de G sur X admet un point fixe.

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Witold MARCISZEWSKI                On some problems concerning Borel structures in function spaces.

Given a space $C(K)$ of continuous real-valued functions on a compact space $K$, we shall consider the following four $\sigma$-algebras in $C(K)$: the cylindrical $\sigma$-algebra $Cyl(C(K))$, i.e., the smallest $\sigma$-algebra, for which all functionals from the dual space $C(K)^*$ are measurable, and the $\sigma$-algebras $Borel(C(K),norm)$, $Borel(C(K), weak)$, $Borel(C(K),pointwise)$ of Borel sets in $C(K)$ with respect to the uniform topology, the weak topology, or the pointwise topology in $C(K)$, respectively. We will discuss some problems concerning these $\sigma$-algebras in $C(K)$. We give an example of a compact space $K$ such that the weak and the pointwise topology generate different Borel structures in $C(K)$.
This is a joint research with Roman Pol.

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