# DST abstracts

### Ando, Hiroshi: Ultraproducts, QWEP von Neumann algebras, and the Effros-Maréchal topology

Haagerup and Winsløw studied the space of von Neumann algebras acting on a separable Hilbert space euiqpped with so-called Effros-Maréchal topology. They proved that this topology is closely linked to the modular theory, tracial ultraproducts and Kirchberg's QWEP conjecture. They in particular showed that a separable type II_1 factor is $R^{\omega}$-embeddable if and only if it is the Effros-Maréchal limit of matrix algebras. In this talk we study further connection among ultraproducts, QWEP and Effros-Maréchal topology. The key ingredients are structural results about non-tracial ultraproducts of von Neumann algebras established last year by the speaker and Haagerup.

This is joint work with Uffe Haagerup and Carl Winsløw.

### Beros, Kostas: Universal subgroups

I will discuss a notion of universality for classes of subgroups of Polish groups. This notion arises from the consideration of a natural Wadge-like pre-order on subgroups of Polish groups.

### Darji, Udayan (Dario): Some examples of universal maps.

We discuss some joint work with E. Matheron concerning universal maps in Banach space theory and topology.

### Dodos, Pandelis: Some recent results in Ramsey Theory

We shall review some results in Ramsey Theory obtained, recently, by the author in collaboration with V. Kanellopoulos, N. Karagiannis and K. Tyros. Among these results are density versions of the classical pigeonhole principles of Halpern-Lauchli and Carlson-Simpson.

### Elekes, Marton: Ranks on Baire class $\alpha$ functions

The well-known theory of ranks on Baire class $1$ functions was developed by Kechris and Louveau. Motivated by some problems related to paradoxic geometric decompositions, we defined various natural ranks on the Baire $\alpha$ classes. To our greatest surprise, it has turned out that all these ranks are bounded below $\omega_1$.

This is joint work with Viktor Kiss.

### Gao, Su: A model for rank one transformations

We define a Polish space for symbolic rank one systems and verify that it is a model for all measure preserving transformations in the sense of Foreman, Rudolph, and Weiss. This in particular implies that symbolic rank one systems can be used to establish any generic dynamical property for all measure preserving transformations.

### Gregoriades, Vassilios: Classes of Polish spaces under effective Borel isomorphism

In this talk we present results about the problem of effective Borel isomorphism between Polish spaces (otherwise $\Delta^1_1$-isomorphism when the spaces are recursively presented). As opposed to the non-effective setting, where only two Polish spaces up to Borel isomorphism exist, the picture in the effective setting is much richer. There exist strictly increasing and strictly decreasing sequences of spaces as well as infinite antichains under the natural notion of effective Borel reduction. In fact this picture occurs in two large categories of spaces, the Kleene spaces and the Spector-Gandy spaces. A key tool for our study is a mapping $T \mapsto \mathcal{N}^T$ from the space of all trees on the naturals to the class of all Polish spaces, for which every recursively presented metric space is $\Delta^1_1$-isomorphic to some $\mathcal{N}^T$ for a recursive $T$, so that the preceding spaces are representatives for the classes of $\Delta^1_1$-isomorphism. Other key tools include the Gandy Basis Theorem and Kreisel compactness. The use of the latter is inspired by a related result of Fokina-Friedman-Törnquist.

### Le Maître, François: Topological generators for full groups

A 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 phenomenons" 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.

### Rosendal, Christian: Large scale geometry of metrisable groups

Large scale geometry of finitely generated or locally compact groups has long been one of the cornerstones of geometric group theory and its connections with harmonic and functional analysis. However, many of the groups of interest in logic, topology and analysis fail to be locally compact, such as automorphism groups of countable structures, diffeomorphism and isometry groups. For these there has been no canonical way of defining their large scale structure, as it is possible, e.g., with the word metric on a finitely generated group. Moreover, recently many groups have turned out to have no non-trivial large scale structure at all, despite being non-compact. We present a theory of large scale structure of metrisable groups and among other things determine the necessary and sufficient conditions for this structure to be unique up to coarse or quasi-isometric equivalences. Applications to model theory will be presented.

### Seward, Brandon: Locally nilpotent groups and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this talk I will discuss a positive solution to this question when the acting group is locally nilpotent. This extends previous results obtained by Gao–Jackson for abelian groups and by Jackson–Kechris–Louveau for finitely generated nilpotent-by-finite groups.

This is joint work with Scott Schneider.

### Slutskyy, Kostyantyn: Automatic continuity for homomorphisms into free products

The main concept of automatic continuity is to establish conditions on topological groups $G$ and $H$ under which any homomorphism from $G$ into $H$ is necessarily continuous. Typically one of the groups is assumed to be very special, while the conditions on the other group are relatively mild.

We shall start with an overview of the automatic continuity results in the situation when the range group is a free group, a free product, or a free product with amalgamation. Of particular interest for us will be homomorphisms $f : G\to A*B$ from a completely metrizable group $G$ into a free product $A*B$ endowed with the discrete topology. The new result to be discussed is that any such homomorphism is necessarily continuous, unless its image is contained in one of the factors.

### Sokic, Miodrag: Semilattices

We consider the class of finite semilattices with respect to the Ramsey property.

### Tsankov, Todor: Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups

It is a well known fact that there is a narrow correspondence between properties of $\omega$-categorical structures and those of their automorphism groups. We investigate properties on the group side that correspond to stability and it turns out that an $\omega$-categorical structure is stable iff every continuous function on the group that is both left and right uniformly continuous is weakly almost periodic. As an application, we show that every such group is minimal, i.e., every continuous surjective homomorphism to another topological group is open, generalizing previous results for particular groups of Stoyanov, Uspenski, Glasner, and others. Our methods can be applied in the setting of both classical and continuous logic.

This is joint work with Itaï Ben Yaacov.