{{Menu ESI|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.