Math 255C, Spring 2009: ``Rigidity in von Neumann Algebras''
Instructor: Sorin Popa
Time and location: MWF 2:00P-2:50P in MS5117

The general theme of this class is the study of rigidity phenomena in von Neumann Algebras of group actions.

Rigidity appears in many areas of mathematics.  Roughly speaking, it occurs whenever
two mathematical objects with rich structure (e.g. groups, actions of groups on spaces)
that are equivalent in a weak sense are shown to be isomorphic as objects with the full
structure.

Some of the most interesting rigidity aspects concerning groups G and their
actions on probability measure spaces, G \actson (X,\mu), are revealed by
the study of the group von Neumann algebras L(G) and the group measure space
von Neumann algebras $L^\infty(X) \rtimes G$, as motivated by the problem of
recovering G (resp. G \actson X) from the isomorphism class of the associated algebra.
Isomorphism of such algebras is a very weak equivalence of groups and group actions.
Rigidity in this context occurs whenever one can establish that an isomorphism of the
von Neumann algebras of certain group actions forces the groups, or the actions,
to share some common properties. An ideal such result recovers the ``whole''
isomorphism class of G (or G \actson X) from its associated von Neumann algebra.

There has been a lot of progress in this area over the last decade. We will explain
some of the main techniques and results obtained thus far, and discuss the main
unsolved problems.

Participating students will be required to make a presentation on assigned material.