Workshop description

For the past decade or more, countable Borel equivalence relations and
actions of countable groups on standard measure spaces has played a central
role in descriptive set theory. At the same time, a veritable explosion of
results in the theory of so-called II_1 factors has taken place in the
field of von Neumann algebras, lead by Sorin Popa and his collaborators.
Interestingly, it is the direct connection between countable,
measure-preserving equivalence relations and certain II_1 factors through
the so-called "group-measure space construction" that has been the driving
engine behind many of the discoveries in von Neumann algebras, and so these
discoveries have undeniable relevance to the concerns of descriptive set
theorists.

The aim of this workshop is to introduce set theorists with
an interest in Borel equivalence relations to von Neumann algebras, and in
particular to the connection between equivalence relations and von Neumann
algebras, and to some of the powerful techniques that the operator
algebraic point of view gives rise to.

After introducing and discussing the basic notions in the morning lectures
of the workshop, the plan is to spend the afternoon discussing some more
advanced topics. Possible topics include (1) applications of descriptive
set theoretic ideas to the classification of II_1 factors, and (2) Popa's
cocycle superrigidity theorems.

Background material and references. The lectures will not require any
familiarity with the theory of von Neumann algebras, but knowledge of the
rudiments of functional analysis (e.g. Hilbert spaces, bounded operators on
these, etc.) will be assumed.

The notion of orbit equivalence for measure preserving actions will play a
central role, but it will not be assumed that course participants are
familiar with this area. Those who want to prepare themselves may want to
look at the following two references.

1. Hjorth, Greg: Countable Borel equivalence relations, Borel reducibility,
and orbit equivalence, Notes available
http://www.math.ucla.edu/~greg/orbitequivalence.pdf

2. Kechris, Alexander S.; Miller, Benjamin D.: Topics in orbit equivalence.
Lecture Notes in Mathematics, 1852. Springer-Verlag, Berlin, 2004.

A gentle introduction to the general theme of the lectures can be found in:

3. Sasyk, Rom?n and T?rnquist, Asger: Borel reducibility and classification
of von Neumann algebras. Bull. Symbolic Logic 15 (2009), no. 2, 169?183.

A less gentle, but still accessible reference is:

4. Popa, Sorin: Deformation and rigidity for group actions and von Neumann
algebras. International Congress of Mathematicians. Vol. I, 445?477, Eur.
Math. Soc., Z?rich, 2007. (Available on the authors web-page.)

A classical general reference in the basics of the field of von Neumann
algebras is:

5. Sakai, Sh?ichir?: $C\sp*$-algebras and $W\sp*$-algebras. Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New
York-Heidelberg, 1971.

Advanced references:

6. Popa, Sorin: On a class of type ${\rm II}_1$ factors with Betti numbers
invariants, Ann. of Math. (2) 163 (2006), no. 3, 809?899.

7. Popa, Sorin:  Cocycle and orbit equivalence superrigidity for malleable
actions of $w$-rigid groups. Invent. Math. 170 (2007), no. 2, 243?295.

8. Popa, Sorin: On the superrigidity of malleable actions with spectral
gap. J. Amer. Math. Soc. 21 (2008), no. 4, 981?1000.

>
>
>