Abstracts
Itai Ben Yaacov
Title: Ample generics in Polish topometric groups
Abstract: A topological group G is said to have ample generics if the action of G on G^n by conjugation admits co-meagre orbits for all n. Polish groups with ample generics and their properties (small index property, automatic continuity and their likes) were studied mostly by Kechris and Rosendal, using results of Hodges, Hodkinson, Lascar and Shelah. Example
are (almost?) exclusively automorphism groups of countable structures, namely, closed subgroups of S_\infty, the permutation group of the naturals.
Motivated by ideas arising from the model theory of metric structures, we suggest that a Polish group may, and should, be considered equipped also with a (non compatible, non separable) bi-invariant metric, e.g., the metric of uniform convergence in the case of the automorphism group
of a metric structure, and that the notion of ample generics should be relativised to this metric (relativising to the discrete 0/1 distance boils down to the classical definition). I shall discuss the definitions, some results (most notably a somewhat surprising variant of the small index property), and a few examples -- automorphism groups of the Hilbert space, the Urysohn sphere and the Lebesgue probability space.
This is joint work with A. Berenstein and J. Melleray.
Valentin Ferenczi
Title: Displaying Polish groups on separable Banach spaces
Abstract: Given a Polish group $G$ and a separable Banach space $X$, when does there exist an equivalent norm on $X$ for which $G$ is topologically isomorphic to the group of linear isometries on $X$? If there exists such a norm, we shall say that $G$ is {\em displayable} on $X$. We shall give characterizations of this property when $G$ is a closed subgroup of $S_{\infty}$, and/or when $X$ is isomorphic to classical spaces such as $c_0$, $l_p$ or $L_p$, and comment on several other results and open problems relative to this question.
This work was done in collaboration with Christian Rosendal.
Matt Foreman
Title: Volume preserving diffeomorphisms of the torus
Abstract: This talk will discuss a recent theorem that shows that the isomorphism relation on volume preserving diffeomorphisms of the torus is not Borel. This is joint work with B. Weiss, extending earlier work with Rudolph and Weiss.
Vladimir Pestov
Title: Ultraproducts of metric groups
Abstract: We will discuss ultraproducts of groups equipped with bi-invariant metrics, and more generally of metric $G$-spaces, and the resulting concepts of hyperlinear groups (Connes, S. Wassermann, Kirchberg, Radulescu) and sofic groups (Gromov, Bengy Weiss, Elek-Szabó), as well as that of a sofic (measure-preserving) equivalence relation, recently introduced by Elek and Lippner. The topics will be put in context of model theory for metric structures as developed by Ben Yaacov, Berenstein, Henson, and Usvyatsov.
Tom Scanlon
Title: Dynamical orbits and o-minimality
Abstract: Skolem introduced (and then many other authors refined and extended) a method to prove instances of Mordell's conjecture whereby it is shown that if $A$ is an abelian variety over the complex numbers, $C \subseteq A$ is an algebraic curve containing the origin and generating $A$ as an algebraic group, and $\Gamma \leq A({\mathbb C})$ is a finitely generated subgroup of the complex points of rank less than
the dimension of $A$, then $C({\mathbb C}) \cap \Gamma$ is finite. The key to Skolem's method is to replace the question about complex algebraic geometry by $p$-adic analysis and then to use the facts that the integers are dense in
${\mathbb Z}_p$ and that a $p$-adic analytic function in one variable which vanishes on infinitely many points must vanish identically on some coset of $p^n {\mathbb Z}_p$ for some
$n \in {\mathbb Z}_+$.
On the face of it, this argument yields no information when ${\mathbb Z}_p$ is replaced by ${\mathbb R}$. However, we show that when analyzing solutions to algebraic equations between points in orbits under discrete dynamical systems near fixed points, in many cases of interest the orbits live in one dimensional sets definable in o-minimal structures and finiteness properties may be deduced from o-minimality.
Asger Tornquist
Title: Global theory of separable C^* algebras and their classification problem
Abstract: I will discuss some recent joint work with Ilijas Farah and Andrew Toms regarding the classification problem for nuclear simple separable C^* algebras. These are the C^* algebras that are the target of the Elliott classification program.
The space of separable C^* algebras has several natural parametrizations as a standard Borel space. We show that the standard constructions in C^* algebras are Borel computable, and in particular that the computation of the "Elliott invariants" are Borel computable. Using results due to Elliott, Thomsen and Villadsen, we show that separable simple nuclear
C^* algebras are not classifiable by countable structures. Finally we discuss possible upper bounds on the complexity of the classification problem for nuclear simple separable C^* algebras, and some open questions.
Todor Tsankov
Title: Unitary representations of oligomorphic groups
Abstract: I will discuss a new classification result for the unitary
representations of oligomorphic permutation groups (or, equivalently, automorphism groups of omega-categorical structures). I will also give some background and mention older similar results about other non-locally compact groups.
Vladimir Uspenskiy
Title: Analytic sets and the diagonal method
Abstract: Goedel's incompleteness theorem, Tarski's undefinability theorem, and the existence of non-Borel analytic sets can all be viewed as applications of Cantor's diagonal method. We'll discuss this point of view, as well as early history of analytic sets, including Lebesgue's error noted by Suslin, the Alexandrov -- Hausdorff theorem on the cardinality of Borel sets, and related developments in topology.