|January 30, 2015|
Thematic Program on Set Theory and Analysis
Workshop on Descriptive Set Theory, Analysis and Dynamical Systems
|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|
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.)
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.
This talk will survey the project of classifying measure preserving
transformations and present some recent "anticlassification"
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.
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
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.
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.
We give Hurewicz-like results concerning Borel subsets of a product of two Polish spaces. This leads to partil uniformigation results
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.
Building on techniques of Andretta and Hjorth, I will present a result which describes how cardinalities (under AD) increase along the Wadge hierarchy.
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.
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.