Appalachian set theory

# Simon Thomas : "Countable Borel equivalence relations"

## Description

This workshop will present an introduction to recent work on countable Borel equivalence relations, a very active area of descriptive set theory which interacts nontrivially with such diverse areas of mathematics as model theory, recursion theory, geometric group theory and ergodic theory.

The lectures will begin with some introductory material on standard Borel spaces and general Borel equivalence relations, including a discussion of some of the many naturally occurring examples. After this, we shall discuss the basic theory of countable Borel equivalence relations, including the remarkable Feldman-Moore Theorem which states that every countable Borel equivalence relation can be realized as the orbit equivalence relation of a Borel action of a suitably chosen countable group. Here a Borel equivalence relation E is said to be countable iff every E-equivalence class is countable. Examples include the Turing equivalence relation on the power set of $\mathbb{N}$, as well as the isomorphism relations on the spaces of finitely generated groups and torsion-free abelian groups of finite rank.

The final lectures will discuss some recent superrigidity theorems of Popa-Ioana and present a number of applications to the theory of countable Borel equivalence relations. (Of course, participants will not be expected to have any prior knowledge of superrigidity theory.)

For those participants who wish to do some background reading before the workshop, an excellent introduction to the subject can be found in:

• A. S. Kechris, New Directions in Descriptive Set Theory, Bull. Symb. Logic 5 (1999), 161-174 [JSTOR]
as well as in the introductory sections of:
• G. Hjorth and A. S. Kechris, Borel equivalence relations and classification of countable models, Ann. Pure Appl. Logic 82 (1996), 221-272 [ScienceDirect]
• S. Jackson, A. S. Kechris and A. Louveau, Countable Borel equivalence relations, J. Math. Logic 2 (2002), 1-80 [WorldSciNet]