May 20, 2018

Thematic Program on Set Theory and Analysis

Workshop on Descriptive Set Theory, Analysis and Dynamical Systems
October 6-12, 2002

I. Farah (Staten Island College),
G. Hjorth (University of California, Los Angeles) and
A.S. Kechris
(California Institute of Technology)

Scot Adams Alain Louveau
Howard Becker M. G. Nadkarni
G. Debs Italy Neeman
Greg Hjorth Vladimir Pestov
Steve Jackson Jean Saint Raymond
Vladimir Kanovei Slawomir Solecki
Robert Kaufman Simon Thomas
Dominique Lecomte Benjamine Weiss

Scot Adams, University of Minnesota
The Origins of Lattice Superrigidity and Dynamical Superrigidity

I will discuss G. Margulis' superrigidity theorem. I will indicate how it arose and how R. Zimmer adapted it to dynamics as part of a larger program initiated by G. Mackey. I will also attempt to explain how this dynamical reformulation found application to the study of Borel reducibility of equivalence relations. (For those who attended the Las Vegas meeting, this will be close to my talk there, but I will try to include some additional detail.)

Howard Becker, University of South Carolina
Polish group actions and generalized model theory

Much of the model theory of countable structures and infinitary logic can be rephrased in group action terms, with respect to the logic actions. We discuss the generalization of these model theoretic concepts and theorems to other actions by other Polish groups.

Matt Foreman, University of California, Irvine
The Classification of measure preserving system

This talk will survey the project of classifying measure preserving transformations and present some recent "anticlassification" results.

Greg Hjorth, University of California, Los Angeles
Treeable equivalence relations and Dye's Theorem

It has been known for some time that any amenable group has only one ergodic action up to orbit equivalence. We will discuss a converse to this result and survey what is known about actioons of the free groups up to orbit equivalence.

Steve Jackson, University of North Texas
On the existence and properties of Steinhaus sets

Answering a question of H. Steinhaus, we show that there is a set in the plane which meets every isometric copy of the integer lattice in exactly one point. We also show that no such set can have the Baire property. We discuss some of the many open problems related to extending these results.

Vladimir Kanovei, Moscow Center for Continuous Mathematical Education
Reducibility of equivalence relations in"hyperfinite" descriptive set theory

As defined by Keisler et al. in 1989, "hyperfinite" descriptiptive set theory studies internal, Borel, projective, and countably determined subsets of hyperfinite sets (in principle, also of *N) in a given nonstandard universe. The results obtained in this area are partially similar to those in the "Polish" DST, partially different, the proofs are usually very different. The talk will concentrate on the reducibility of Borel and countably determined equivalence relations in nonstandard domain.

Robert Kaufman,University of Illinois at Urbana-Champaign
Mixing and Descriptive Set Theory

Let X be a compact metric space and H(X) the metric space of continuous selfmaps of X. The subset H(X,m) is then defined as follows: A transformation T belongs to H(X,m) provided there is a T-invariant probability measue mu such that T is mixing for the measure mu.
Example (S.Siboni) For a certain space X, H(m) in't closed.
Theorem 1 H(m) is always an analytic set.
Theorem 2 For a certain space Y, H(m) is a complete analytic
subset of H.
The space Y is immense, but further effort yields an example in which X is a Cantor set and Y has dimenion 1.

Dominique Lecomte, Universite Paris VI
About the complexity of Boreal subsets of the plane

We give Hurewicz-like results concerning Borel subsets of a product of two Polish spaces. This leads to partil uniformigation results

Alain Louveau, Universite Paris VI
Complete analytic equivalence relations and orders

I will present some results about existence and non-existence of complete (i.e., maximum in the Borel reducibility ordering) elements in some classes of analytic or Borel relations : equivalence relations, quasi-orders, partial orders. Some of the results are joint with Christian Rosendal.

Italy Neeman, University of California at Los Angeles
Effective cardinalities along the Wadge Hierarchy

Building on techniques of Andretta and Hjorth, I will present a result which describes how cardinalities (under AD) increase along the Wadge hierarchy.

Slawomir Solecki, University of Illinois at Urbana-Champaign
Measuring subsets of discrete groups and Haar null sets

I will introduce and discuss natural ways of assessing size of subsets of discrete groups. Some traces of these notions can be found in the work of Mitchell, Day, and Christensen. I will show that the relationships between the various ways of measuring subsets of a group depend heavily on algebraic properties of the group. Amenable, infinite conjugacy classes, and finite conjugacy classes groups will be relevant. I will present applications of these results to Haar null subsets of Polish group.

Benjamin Weiss, Hebrew University of Jerusalem
Generic Dynamics - an update

Generic dynamics studies those dynamical properties of continuous actions that are valid modulo the ideal of first category sets. After explaining the basic concepts and results I will survey some of the newer developments in this area.

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