A topological group G is said to have ample generics if the action of G on Gⁿ 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_∞, 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 space and the Lebesgue probability space. This is joint work with A. Berenstein and J. Melleray.

Borel reducibility is a tool from descriptive set theory which allows one to compare the complexity of classification problems from algebra, analysis and logic. Since many classification problems are captured by equivalence relations induced by a natural group action, this area sometimes resembles orbit equivalence theory. In this talk, we will consider the relation induced by the action of SL_n(Z) on SL_n(Z_p) (the p-adics). Motivated by a connection with torsion-free abelian groups, Hjorth-Thomas essentially showed that as p varies, the corresponding relations are incomparable with respect to Borel reducibility. The proof requires powerful tools: either Zimmer's superrigidity theorem for lattices in Lie groups, or Ioana's recent superrigidity theorem for profinite actions. I'll indicate how this works, and time permitting, give some newer incomparability results.

Uniformly HyperFinite (UHF) algebras are C*-algebras in which every finite subset is 'near' a finite-dimensional full matrix subalgebra. This can be formalized in three different ways, all three being equivalent in the separable case. Separable UHF algebras were classified in the 1960s by Glimm and Dixmier. Dixmier asked whether the three definitions are equivalent in the nonseparable case. I will give a complete answer to this question. One of the methods used is a new way of associating a C*-algebra to a graph. I will also address the question of the homogeneity of the pure state space of simple C*-algebras, answered in the separable case by Kishimoto-Ozawa-Sakai. A part of this work is joint with Takeshi Katsura.

Gurarii's space is the unique separable existentially closed Banach space. He proved existence in the 1960s; Lusky proved uniqueness in the 1970s, making substantial use of structural results about L₁-preduals proved by Lindenstrauss and others. This talk will focus on how Gurarii's space looks from the point of view of model theory (specifically, model theory of metric structures, based on the continuous version of first-order logic). Some rather general tools from model theory provide proofs of existence and uniqueness that are geometrically simpler than the original ones. They also provide some insight into open questions, such as: how complex is the action of the automorphism group of Gurarii's space on its unit sphere?

Descriptive set theory provides a means for analyzing the complexity of classification problems through the notion of Borel reducibility. I will survey the problem of classification in ergodic theory from this perspective, and in particular discuss a result of Foreman and Weiss which applies Hjorth's notion of turbulence to the study of measure-preserving actions of amenable groups.

I will describe recent joint work with R. Lyons and S. Vassout in which we relate isoperimetric type constants for measurable equivalence relations to Gaboriau's cost invariant.

We shall present the two central games for block sequences in Banach spaces: the Gowers game and the infinite asymptotic game, and show how these can be used to give a transparent proof of a basic Ramsey principle of Gowers. With this, we shall present a number of dichotomies for Banach spaces laying the groundwork for a coarse classification programme.

The talk is centered around the conjecture that (for a reasonable class of Banach spaces) the asymptotic structure is preserved under uniform homeomorphisms. We will present some evidence (both old and new) towards this goal.

Classical model theory has had little application to functional analysis, notwithstanding the fact that standard objects such as Banach spaces, C*-algebras, and tracial von Neumann algebras carry useful notions of ultraproduct. Recently, building on many years' development, it was realized that an elegant model theory for these structures can be based on a logic in which truth values lie in the interval [0,1]. This "continuous model theory" recovers analogues of many classical logical theorems involving ultraproducts, making new tools available to analysts. Recent work along these lines has answered some published questions and raised new ones. I will try to give a "big picture" survey of the subject, including history and basic background for both continuous model theory and operator algebras.

It has been shown (Losert, Neufang, Pachl, S.) that the measure algebra of every locally compact group is Arens irregular &mdash that is to say, the topological centre of the double dual is trivial. Parts of this argument, along with relevant definitions, will be discussed highlighting the role of factorization lemmas. Set theoretic aspects of these arguments will be examined.

The classification problem for nuclear simple separable C*-algebras is a problem of central importance in current research in C*-algebras. In this talk I will discuss some recent joint work with Ilijas Farah and Andrew Toms in which we study this classification problem from the point of view of descriptive set theory and Borel reducibility. We show that nuclear simple separable C*-algebras cannot be classified by countable structures, that isomorphism is an analytic equivalence relation which is not Borel, and that it is strictly below the universal analytic equivalence relation in the Borel reducibility hierarchy. Interestingly, the results rely on functorial classification results in C*-algebra theory.

Harmonic analysis has traditionally been developed on locally compact groups, where the existence of the Haar measure and the regular representation provide the basis of the theory. The Haar measure is not available on non-locally compact groups but it is sometimes still possible to develop a reasonable representation theory using different methods. In this series of lectures, I will mostly concentrate on closed permutation groups of a countably infinite set (or automorphism groups of countable structures), a class of groups studied both in permutation group theory and model theory.