Rigidity for von Neumann algebras and their invariants

We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with uncountable fundamental groups and the construction of W*-superrigid actions where the crossed product entirely remembers the initial group action that it was constructed from.Comment: ICM 2010 Proceedings tex

Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \C^k into \C^n ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of \C^n can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \C^n we derive the existence of families of holomorphic \C^*-actions on \C^n ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of \C^* actions on \C^n (with prescribed linear part at a fixed point)

Problems on Polytopes, Their Groups, and Realizations

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete Geometry, to appear