# 6th Descriptive Set Theory in Paris, December 16 – 17, 2013

The sixth meeting Descriptive Set Theory in Paris  organized by the Equipe d’Analyse Fonctionnelle will be held on December 16 and 17, 2013 at the Institut de Mathématique de Jussieu.

## List Of Speakers

• Itai BEN YAACOV
• John CLEMENS
• Sy FRIEDMAN
• David KERR
• Aleksandra KWIATKOWSKA
• François LE MAITRE
• Luca MOTTO ROS
• Lionel NGUYEN VAN THE
• Marcin SABOK
• Omar SELIM
• Konstantin SLUTSKY
• Slawomir SOLECKI
• Asger TORNQUIST
• Anatoly VERSHIK

## Titles and Abstracts

Itai BEN YAACOV                Model theory and Roelcke pre-compact groupsIn a recent work with Todor TSANKOV we have used « model-theoretic reasoning » to prove facts regarding Roelcke precompact Polish groups, e.g., that if every uniformly continuous function on $G$ is wap then $G$ is totally minimal. I shall attempt to present the model-theoretic point of view for this and possibly other recent results of a similar nature.

John CLEMENS                Relative primeness of equivalence relationsLet $E$ and $F$ be equivalence relations on the spaces $X$ and $Y$. We say that $E$ is prime to $F$ if: whenever $\varphi: X \rightarrow Y$ is a homomorphism from $E$ to $F$, there is a continuous embedding $\rho$ from $E$ to itself so that the range of $\varphi \circ \rho$ is contained in a single $F$ class. That is to say, $\varphi$ is constant (up to $F$-equivalence) on a set on which $E$ maintains its full complexity with respect to Borel reducibility. I will discuss this notion and show that many non-reducibility results in the theory of Borel equivalence relations can be strengthened to produce primeness results. I will also discuss the possibility of new types of dichotomies involving the notion of primeness.

Sy FRIEDMAN                The Silver Dichotomy in Generalised Baire SpaceI’ll discuss Silver’s Dichotomy for Borel equivalence relations in the space $\kappa^\kappa$ where $\kappa$ is regular and uncountable (and there are only $\kappa$ bounded subsets of $\kappa$). The dichotomy fails badly in L. The main result is that assuming the existence of $0\sharp$ there is a forcing extension of L in which it holds.

Vassilis GREGORIADES                A recursive theoretic view to the decomposability conjectureThe decomposability conjecture states that every function from an analytic space to a separable metric space, for which the preimage of a $\Sigma^0_{m+1}$ set is a $\Sigma^0_{n+1}$ set, where $1 \leq m \leq n$, is decomposable into countably many $\Sigma^0_{n-m+1}$-measurable functions on $\Pi^0_n$ domains. In this talk we present recent results about this problem based on work of Kihara and the speaker. The key tools for the proofs come from recursion theory and effective descriptive set theory, and include a Lemma by Kihara on canceling out Turing jumps and Louveau separation.

Adriane KAÏCHOUH                Amenability and Ramsey theory in the metric settingMoore recently characterized the amenability of closed subgroups of $S_\infty$ by a Ramsey-type property (in the vein of the Kechris-Pestov-Todorcevic correspondence). We will present this property and give a generalization of Moore’s result to general Polish groups. We will also discuss some nice consequences of this characterization.

David KERR                Borel complexity and automorphisms of operator algebrasWe show that, for the standard examples of strongly self-absorbing $C^*$-algebras, the action of the automorphism group on itself by conjugation is generically turbulent. Using this we then prove that, for all separable $C^*$-algebras which tensorially absorb the Jiang-Su algebra, the relation of conjugacy on automorphisms is not classifiable by countable structures. This includes the $C^*$-algebras that fall under the scope of classification results based on the Elliott invariant. The main novelty of our turbulence argument is the use of the malleability of the tensor product shift over a strongly self-absorbing $C^*$-algebra. This is joint work with M. Lupini, N.C. Phillips, and W. Winter.

Aleksandra KWIATKOWSKA                Lelek fan from a projective Fraisse limitThe Lelek fan is the unique compact, connected subspace of the cone over the Cantor set that has a dense set of endpoints. We show how to obtain the Lelek fan as a quotient of a projective Fraisse limit of a family of finite rooted trees. Using this construction, we show projective universality and projective homogeneity of the Lelek fan. We further show that the homeomorphism group of the Lelek fan is totally disconnected, generated by every neighbourhood of the identity, and has a dense conjugacy class. This is a joint work with Dana Bartosova.

François LE MAITRE                 Topological rank for full groupsA theorem of Dye asserts that two full groups of ergodic pmp equivalence relations are isomorphic iff the equivalence relations are orbit equivalent, so that the study of full groups might provide new invariants for pmp equivalence relations. Here we will focus on the topological rank of the full group, that is, the minimal number of elements needed to generate a dense subgroup. We will also discuss “genericity phenomena” for topological generators, motivated by the Schreier-Ulam theorem which states that the generic pair in a compact metrisable connected group does generate a dense subgroup. If time permits, some results on non ergodic equivalence relations will be mentioned.

Luca MOTTO ROS                Lipschitz and uniformly continuous reducibilitiesWe analyze the reducibilities induced by, respectively, uniformly continuous, Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces, and give sufficient and necessary (topological) conditions on such spaces for the corresponding degree-structures being well-behaved. This is joint work with P. Schlicht.

Lionel NGUYEN VAN THE                Universal minimal proximal flows of non-Archimedean Polish groupsGiven a topological group $G$, certain classes of minimal $G$-flows admit a unique universal element. Proximal flows fall into that category, and the purpose of this talk will be to describe explicitly the universal object attached to various non-Archimedean Polish groups.

Marcin SABOK                Completeness of the isomorphism problem for separable $C^*$-algebrasI will discuss a recent result which says that the isomorphism problem for separable nuclear $C^*$-algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI $C^*$-algebras is a complete orbit equivalence relation. This means that any isomorphism problem arising from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple, separable AI $C^*$-algebras. As a consequence, we get that the isomorphism problems for separable nuclear C*-algebras and for separable $C^*$-algebras have the same complexity. This answers questions posed by Elliott, Farah, Paulsen, Rosendal, Toms and Törnquist.

Omar SELIM                An alternative proof of Choquet’s theoremWe give an alternative proof of Choquet’s theorem concerning capacities of infinite order.

Konstantin SLUTSKY                Regular cross-sections of Borel flows.

A theorem of Ambrose and Kakutani asserts that up to a null set any free measure-preserving flow on a standard Lebesgue space is isomorphic to a “flow under a function.” Rudolph showed that one can always assume that it is a flow under a function which attains just two values. Wagh managed to prove the analogue of Ambrose-Kakutani theorem in a purely Borel context: any free Borel flow on a standard Borel space is isomorphic to a flow under a function. Nadkarni asked whether Rudolph’s result holds true in this more general setting. We answer Nadkarni’s question in the affirmative and prove that any free Borel flow is indeed a flow under a two-valued function.

Slawomir SOLECKI                Dual Ramsey theorem for trees and its contextThe classical Ramsey theorem was generalized in two major ways: to the dual Ramsey theorem by Graham and Rothschild and to Ramsey theorems for trees initially by Deuber and Leeb. Bringing these two lines of thought together, I will present the dual Ramsey theorem for trees and its context. Galois connections are important in formulating this theorem. The abstract approach to Ramsey theory, I developed earlier, is used in its proof.

Asger TORNQUIST                Mad families and the Solovay model.In this talk I will discuss a new proof of a theorem of Mathias’ which states that there are no analytic infinite mad families. Then I will show that this proof can be adapted to show that if (1) All sets have the perfect set property and (2) $OCA_\infty$, a variant of the open colouring axiom, holds then there are no infinite mad families. In particular, there are no infinite mad families in Solovay’s model.

Anatoly VERSHIK                Infinite tower of the measures and its applicationsThe construction of the tower of the standard measures will be described. Besides its interest itself there are applications to the classification of the measurable sets with respect to several groups of symmetries of measure preserving transformations.