Cocycle Superrigidity and Equivalence Relations
Let X be a standard Borel space. Given two countable equivalence relations R and S on X, we write R <= S if there is a Borel f : X --> X such that the f-preimage of S is R. It was a longstanding open question whether <= is a non-linear ordering, i.e., whether there exist R and S such that neither R <= S nor S <= R is true. In 1998, A. Kechris observed this question could be answered given sufficient knowledge of measurable cocycles on measure-preserving equivalence relations on finite measure spaces. The needed information was provided by R. Zimmer's cocycle superrigidity theorem, a generalization of G. Margulis' homomorphism superrigidity theorem. We will talk about superrigidity, and how it applies to the ordering <= on equivalence relations.

Howard Becker
University of South Carolina

Cocycles and continuity
Mackey's Cocycle Theorem (every Borel almost cocycle is equivalent to a Borel strict cocycle) is a result about locally compact Polish groups which is not true for arbitrary Polish groups. I will discuss this theorem, the open problem of its generalization to some non-locally compact groups, and the origin of this problem in the foundations of quantum mechanics. Historically, research on this topic by Mackey and others has been in the context of standard Borel G-spaces. Now that it has been proved by Kechris and others that these spaces have topological realizations as Polish G-spaces, this topic can be studied from a topological point of view. The property of Mackey's Theorem turns out to be equivalent to continuity properties of the almost cocycle.

Greg Hjorth
UCLA

Alain Louveau
University of Paris

Maharam's problem
In this talk, I will sketch the construction, due to Talagrand,
of a Maharam submeasure on the Cantor space which is orthogonal to all
measures, thus solving the long-standing problem known as Maharam's, or as
the Control Measure Problem.

Asymptotic Enumeration of Spanning Trees via Traces and Random Walks
Methods of enumeration of spanning trees in a finite graph and relations to various areas of mathematics and physics have been investigated for more than 150 years. We will review the history and applications. Then we will give new formulas for the asymptotics of the number of spanning trees of a graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call "tree entropy", which we show is a logarithm of a normalized [Fuglede-Kadison] determinant of the graph Laplacian for infinite graphs. Proofs involve new traces and the theory of random walks.

Yiannis Moschovakis
UCLA

Recursion and complexity
How many integer divisions must you do to decide whether two arbitrary n-bit numbers are coprime? Can you do better than the euclidean, which requires no more than 2n divisions? And could you do (asymptotically) even better, if you were allowed to use a different algorithm for each n?

One aim of this talk is to explain how the representation of algorithms from given primitives by recursive programs can be used to derive very widely applicable lower bounds for the complexity of basic decision problems in arithmetic. I will use as examples some of the results in [1] and [2] below, which I will outline briefly; but the emphasis in this talk will be on the problem of distinguishing between uniform and non-uniform algorithms. This work is joint with Lou van den Dries.

REFERENCES
[1] Lou van den Dries and Yiannis N. Moschovakis. Is the Euclidean algorithm optimal among its peers? The Bulletin of Symbolic Logic, 10:390-418, 2004.
[2] Lou van den Dries and Yiannis N. Moschovakis. Arithmetic complexity. In preparation.

Sorin Popa
UCLA

On the superrigidity of malleable actions
The property (T) assumption on a group G is well known to automatically entail a variety of rigidity properties for all measure preserving actions of G on [0,1]. I will present a series of results showing that a certain malleability (+ weak mixing) condition on the way G acts on [0,1] enhances considerably the overall rigidity of the group-action. For instance, if some free action of an arbitrary group has same orbits (a.e.) as the orbits of a malleable action of a property (T) group with no finite normal subgroups, then the two actions must be conjugate. Generalized Bernoulli actions and Gaussian actions are typical examples of malleable actions.

Superrigidity and Classification Problems
Suppose that is a class of countable structures. In this talk, following Friedman-Stanley and Hjorth-Kechris, I will explain how to use the theory of Borel equivalence relations to analyze the isomorphism relation on and develop a framework for measuring the complexity of possible complete invariants. Throughout, I will concentrate on classes of structures which are "finitely generated" in some broad sense, such as the class of finitely generated groups, the class of torsion-free abelian groups of finite rank, the class of fields of finite transcendence dimension, etc. These are the classes whose isomorphism relations correspond to countable Borel equivalence relations; i.e. those Borel equivalence relations E such that every E-equivalence class is countable. As we shall see, this means that the superrigidity results of Zimmer, Furman and Popa can (occasionally) be applied.

Stevo Todorcevic
University of Toronto/University of Paris

The Separable Quotient Problem and its Relatives
We discuss some old and new results about the problem of finding a quotient of a given Banach space with a sufficiently long Schauder basis. We also discuss some of the related problems.

Beyond w-huge
If there is an inner model theory for one supercompact cardinal then essentially all known large cardinal axioms are absolute between V and the inner model for one supercompact cardinal. This must hold no matter how the inner model is constructed. The initial analysis has revealed a new hierarchy of large cardinal axioms. These axioms lie in a realm beyond w-huge and are in striking parallel with determinacy axioms. If there is an inner model theory for one supercompact cardinal then these new axioms are in fact intimately connected with determinacy axioms.