Smooth classification of actions of higher rank groups (with hyperbolicity) has been a topic of interest for few decades. Many differentiable rigidity properties of the irreducible algebraic models support the following conjecture made by A. Katok and R. Spatzier: All ''irreducible" smooth Anosov Z^k actions on any compact smooth manifold are smooth conjugate to algebraic models. Katok-Spatzier conjecture is proved under the assumption the supporting manifold is an infranilmanifold. In a more general problem, i.e. partially hyperbolic abelian actions of higher-rank abelian groups there have been so far no smooth classification results.

In this talk I will talk about our recent progress towards Katok-Spatzier conjecture on general manifold (joint work with D. Damjanovic). And for partially hyperbolic action with compact center fibers, (joint work with D. Damjanovic and A. Wilkinson, working in progress) we also get a global classification result if the center is one or two dimensional. Speaker(s): Disheng Xu (University of Chicago)