
THEMATIC PROGRAMS 

July 7, 2015  
Thematic Program on Set Theory and AnalysisWorkshop on Descriptive Set Theory, Analysis and Dynamical Systems

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 UrbanaChampaign
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 Tinvariant 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 Hurewiczlike 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 nonexistence of complete (i.e., maximum in the Borel reducibility ordering) elements in some classes of analytic or Borel relations : equivalence relations, quasiorders, 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 UrbanaChampaign
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.