# DESCRIPTIVE SET THEORY IN PARIS

## Titles and Abstracts of the talks

 Itai BEN YAACOV Model-theoretic isolation and smooth vectors in the Gurarij space John CLEMENS Injective selectors and pointed trees Pandelis DODOS New results in Ramsey Theory for trees Mirna DZAMONJA A Hausdorff-like characterization of FAC posets Marton ELEKES Reconstructing geometric objects from the measures of their intersections with test sets Eli GLASNER Minimal Hyperspace actions of homeomorphism groups of H-homogeneous spaces Gilles GODEFROY Tightness of Banach spaces and Baire category. Stephen JACKSON Colorings, Supercolorings, and the dynamics of group actions. Julien MELLERAY Extensions of measure-preserving actions Benjamin MILLER Incomparable treeable equivalence relations Luca MOTTO ROS $\kappa$-Souslin quasi-orders and definable cardinality Lionel NGUYEN VAN THÉ More on the Kechris-Pestov-Todorcevic correspondance: precompact expansions Philipp SCHLICHT Continuous reducibility for the real line Victor SELIVANOV $\Delta^0_\alpha$-Reductions in Quasi-Polish Spaces Konstantin SLUTSKY Free products and HNN extensions of groups with two-sided invariant metrics Vladimir USPENSKIY On some problems in topological dynamics

Itai BEN YAACOV               Model-theoretic isolation and smooth vectors in the Gurarij space

Just like the Urysohn space can be described as the unique separable homogeneous and universal metric space, the (somewhat less familiar) Gurarij space can be described as the unique separable, universal and approximately homogeneous Banach space -- that is to say that an isometric isomorphism between two finite-dimensional subspaces is arbitrarily close to one which extends to an isometric automorphism, but need not extend to an automorphism itself. The uniqueness of the Gurarij space, whose initial definition was not quite identical to the one given here, was first proved by Lusky in 1976, who, by a " modify the previous proof " argument, also proves that the set of smooth points on the unit ball of the Gurarij space form an orbit under isometric automorphism. We propose to use the vocabulary, and some of the machinery, of metric model theory (specifically: isolation of types and atomic models), to give a more conceptual proof of the same two results, as well as of generalisations thereof. No prior familiarity with model theory will be required.

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John CLEMENS               Injective selectors and pointed trees

I will discuss effective versions of a measure-theoretic selection theorem of Graf and Mauldin, and applications to uniformly branching trees. In particular, an effectively random tree contains a branch which can compute the tree, but an effectively generic tree does not.

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Pandelis DODOS               New results in Ramsey Theory for trees

We shall review some recent advances in Ramsey Theory for trees focusing in particular on the Halpern-Lauchli Theorem. The Halpern-Lauchli Theorem, discovered in 1966, is a deep pigeon-hole principle for trees and concerns partitions of the level product of a finite sequence of uniquely rooted, finitely branching trees without maximal nodes. It has been the main tool for the development of Ramsey Theory for trees, a rich area of Combinatorics with significant applications, most notably in the Geometry of Banach spaces. A density version of the Halpern-Lauchli Theorem was conjectured by R. Laver in the late 1960s and obtained recently (2010-2011). We shall discuss both the infinite and the finite version of the density Halpern-Lauchli Theorem, as well as, the bounds we get from the argument. We shall also present some consequences who are pointing towards a "random" version of Milliken's Theorem for strong subtrees. This is joint work with V. Kanellopoulos, N. Karagiannis and K. Tyros.

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Mirna DZAMONJA               A Hausdorff-like characterization of FAC posets

We shall present the following theorem, obtained jointly with U.Abraham, R.Bonnet, J.Cummings and K. Thompson (to appear in TAMS) Theorem Let BP be the class of posets P such that P is either a wqo poset, the dual of a wqo poset, or a linear ordering. Let P be the smallest class of posets such that (a) P contains BP; (b) P is closed under lexicographic sums with index set in BP; (c) P is closed under augmentation. Then P is the class of posets with no infinite antichains. We shall also discuss a recent application of this theorem in the work of Duffus, Laflamme, Pouzet and Woodrow.

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Marton ELEKES               Reconstructing geometric objects from the measures of their intersections with test sets

We say that an element of a given family $\mathcal{A}$ of subsets of $\mathbb{R}^d$ can be reconstructed using $n$ test sets if there exist $T_1,\ldots,T_n \subset \mathbb{R}^d$ such that whenever $A,B \in \mathcal{A}$ and the Lebesgue measures of $A \cap T_i$ and $B \cap T_i$ agree for each $i=1,\ldots,n$ then $A=B$. Our goal will be to find the least such $n$. If $\mathcal{A}$ consists of the translates of a fixed reasonably nice subset of $\mathbb{R}^d$ then this minimum is $n=d$. In order to obtain this result, on the one hand we have to reconstruct a translate of a fixed absolutely continuous function of one variable using $1$ test set. On the other hand, we need that under rather mild conditions the Radon transform of the characteristic function of $K$ (that is, the measure function of the sections of $K$), $(R_\theta \chi_K) (r) = \lambda^{d-1} (K \cap \{x \in \mathbb{R}^d : \langle x,\theta \rangle = r \})$ is absolutely continuous modulo a nullset for almost every direction $\theta$. These results are based on techniques of Fourier analysis. If $\mathcal{A}$ consists of the magnified copies $rE+t$ $(r \ge 1, t \in \mathbb{R}^d)$ of a fixed reasonably nice set $E \subset \mathbb {R}^d$, where $d \ge 2$, then $d+1$ test sets reconstruct an element of $\mathcal{A}$. This fails in $\mathbb{R}$: an interval, and even an interval of length at least $1$ cannot be reconstructed using $2$ test sets. Finally, an element of a reasonably nice $k$-dimensional family of geometric objects can be reconstructed using $2k+1$ test sets. These test sets are obtained by a random construction. A example from algebraic topology shows that $2k+1$ is sharp in general. }

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Eli GLASNER               Minimal Hyperspace actions of homeomorphism groups of H-homogeneous spaces

Let $X$ be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. $X$ is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call \emph{stable collections} of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system $(Exp(Exp(X)),Homeo(X))$, where $Exp(X)$ stands for the hyperspace comprising the closed subsets of $X$ equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij's characterization of the universal ambit of $G=Homeo(X)$. The results apply to $X=C$ the Cantor set, the generalized Cantor sets $X=\{0,1\}^{\kappa}$ for non-countable cardinals $\kappa$, and to several other spaces. A particular interesting case is $X=\omega^{\ast}=\beta\omega\setminus\omega$, where $\beta\omega$ denotes the Stone-\v Cech compactification of the natural numbers. This is a joint work with Yonatan Gutman.

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Gilles GODEFROY               Tightness of Banach spaces and Baire category.

In this joint work with V. Ferenczi, we show for any Banach space $X$ with a basis, the relations of isomorphism and bi-embedding are meager or co-meager on the Polish set of block-subspaces of $X$. We relate this result with tightness and minimality of Banach spaces.

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Stephen JACKSON               Colorings, Supercolorings, and the dynamics of group actions.

We introduce the notions of colorings and supercolorings on countable groups and relate them to the dynamics of the group actions. This leads to new questions and results about the marker structures on groups, and new connections between the dynamics of the group actions and the group combinatorics.

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Julien MELLERAY               Extensions of measure-preserving actions

In 2003, Ageev announced the following result: if G is an infinite cyclic group of a countable abelian group H, then a generic measure-preserving action of G (on a standard probability space) can be extended to a free action of H. In this talk I will explain how one can extend this result to the case when G is any finitely generated abelian group, and discuss some possible generalizations of this theorem.

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Benjamin MILLER               Incomparable treeable equivalence relations

I will discuss generalizations, simplifications, and new applications of Hjorth's results on incomparable treeable equivalence relations.

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Luca MOTTO ROS               $\kappa$-Souslin quasi-orders and definable cardinality

A quasi-order $S$ on a Polish space is called $\kappa$-Souslin if it is a continuous image of the space $\kappa^\omega$ endowed with the product of the discrete topologies on $\kappa$. Extending previous works of Louveau-Rosendal and Friedman-Motto Ros on the case $\kappa = \omega$, we show (in ZF + DC alone) that the embeddability relation between combinatorial trees (i.e. connected acyclic graphs) of size $\kappa$ is complete for the class of $\kappa$-Souslin quasi-orders, i.e. that every $\kappa$-Souslin quasi-order is reducible to the mentioned embeddability relation. Moreover, we show that for every $\kappa$-Souslin quasi-order $S$ there is an $L_{\kappa^+ \kappa}$-sentence $\phi$ such that $S$ is bi-reducible with the embeddability relation on models of $\phi$ (where all reductions involved are of low'' topological complexity and are explicitly definable from any continuous function witnessing the $\kappa$-Sousliness of $S$). The mentioned results can be applied to get nice corollaries both in models with choice (like the Solovay's model) and in models of suitable determinacy axioms, and have also consequences concerning definable cardinality. This is joint work with A. Andretta.

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Lionel NGUYEN VAN THÉ               More on the Kechris-Pestov-Todorcevic correspondance: precompact expansions

In 2005, the paper "Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups" by Kechris, Pestov and Todorcevic provided a poverful tool to compute universal minimal flows. This immediately led to an explicit representation of this invariant in many concrete cases. However, as more and more cases were investigated, it became apparent that in some particular situations, the framework does not allow to perform the computation directly, but only after a slight modification of the original argument. The purpose of the talk will be to present a general framework which allows to avoid that twist and is well suited for further applications.

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Philipp SCHLICHT               Continuous reducibility for the real line

We analyze the quasi-order of continuous reduction on the set of Borel subsets of the real line. We show that the power set of the natural numbers with almost inclusion embeds into this quasi-order. We also consider other Polish spaces. This is joint work with Daisuke Ikegami.

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Victor SELIVANOV               $\Delta^0_\alpha$-Reductions in Quasi-Polish Spaces

There are several directions to generalize the classical Wadge reducibility on the Baire space, in particular to a wider class of reducing functions or to more complicated topological spaces. For a space X and a poinclass $C\subseteq P(X)$, C-reducibility is the preorder on $P(X)$ corresponding to many-one reductions by functions on X such that the preimage of any set in C is again in C. In a series of papers. A. Andretta, D. Martin and L.M. Ros have shown that, under suitable set-theoretic assumptions, the structure of C-degrees in the Baire space is isormorphic to the structure of Wadge degrees, where C is the class of Borel sets or is a level of the Borel hierarchy. P. Hertling has shown that the structure of Wadge degrees in the space of reals is much more complicated than the structure of Wadge degrees in the Baire space. We show that for many C-reducibilities (this applies e.g. to the case when C is any infinite level of the Borel hierarchy) the structure of C-degrees in any uncountable quasi-Polish space X is isormorphic to the structure of Wadge degrees in the Baire space. This immediately follows from the following extension and refinement of a classical fact: any two uncountable quasi-Polish spaces X,Y are C-isomorphic, where C(X) is the class of sets of finite Borel rank in X, and C-isomorhism is a bijection between X and Y which preserves the classes C(X) and C(Y) in both directions. Quasi-Polish spaces is a natural class of spaces (countably-based completely quasi-metrizable spaces) containing all Polish spaces and all omega-continuous domains.

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Konstantin SLUTSKY               Free products and HNN extensions of groups with two-sided invariant metrics

M. Graev in the 40s constructed special two-sided invariant metrics on free groups F(X) starting with a metric on the basis X. We will discuss similar constructions for free products (possibly with amalgamation) of groups with two-sided invariant metrics and for HNN extensions of such groups.

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Vladimir USPENSKIY               On some problems in topological dynamics

By a dynamical system we mean a compact space X together with a continuous action of a group G. The enveloping semigroup of such a system is the closure of the image of G in the compact space X^X of all self-maps of X. We'll characterize those dynamical systems for which the enveloping semigroup is metrizable (joint work with Glasner and Megrelishvili). One approach uses Baire 1 functions, another is based on representations of groups by isometries of Banach spaces. We'll also discuss some open problems, including Glasner's problem: if T is a fixed-point-free self-homeomorphism of a compact space X, does it follow that the group generated by T, equipped with the uniform convergence topology, admits a non-trivial continuous character?

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