Mathematical Sciences: Analytic and Geometric Questions for Smooth Group Actions

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Abstract Feres The main purpose of the project is to explore a number of connections between Differential Geometry and Dynamical Systems associated to actions of Lie groups and foliations. Particular attention is given to smooth actions of semisimple Lie groups of real rank at least 2 and their lattices. Other problems considered are concerned with representations of amenable groupoids and the method of Transference, and foliations with Kazhdan's property T. The general theory of Dynamical Systems provides tools used to understand the long term development of processes, whether they are physical, biological, or purely mathematical. Ordinarily, the process evolution is considered as a function of a single parameter, usually interpreted as time. However, some qualitatively new phenomena arise in multiparameter systems. Of particular concern in this project is a kind of ''rigidity'' property. (The parameter space is represented by the mathematical notions of a ''group'' or a ''foliation.'' Some of the ''groups'' considered here, namely lattice groups with ''property T,'' have surprising connections with such a subject as optimal connectivity properties of information networks). The basic aim of this ongoing project is to explore a few of the more basic and fundamental questions of the subject by means of geometric and probabilistic techniques.