Rigidity of group actions in Ergodic Theory, and related topics

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The project focuses on three areas of rigidity in the theory of "large" groups and in Ergodic theory: (1) extending Margulis' superrigidity and Zimmer's cocycle superrigidity from Lie groups setting to other situations, including A_2 groups and lattices in products;(2) exploring the connections between recent advances due to Popa in von Neumann Algebras in the context of Ergodic theory;(3) studying measure rigidity questions in the spirit of Ratner's theorem for algebraic actions of non-amenable groups (which have no unipotents).The common phenomenon, termed "rigidity", in the above problems is that certain algebraic structures describing intrinsic symmetries of the studied objects impose very strong constrains on the otherwise flexible characteristics of these objects. One of the most intriguing aspects of what became known as "rigidity theory" is its characteristic synergy of methods and tools from rather different areas: Algebra, Number theory, Geometry and Analysis. This project concerns some problems in Dynamics most related to Lie groups and Analysis.