Mathematical Sciences: Ergodic Theory, Groups, and Geometry

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Professor Zimmer will study a variety of questions within the general program of understanding the actions of semisimple Lie groups and their discrete subgroups on manifolds. This includes an investigation of the topological, geometric, and analytic properties of manifolds admitting such an action, and the measure theoretic, topological, and geometric structure of the actions themselves. The techniques of ergodic theory and algebraic groups, combined with the geometric structure inherent in the situation, will be essential throughout the work. This award will also support three junior investigators who will conduct research in the same general area. Groups usually arise as transformations which preserve the symmetry of some system much as the group of rotations preserves the symmetry of a sphere. Given an abstract group there are often many ways it can be realized or represented as a group of transformations. There has been much research done on the representation of groups as linear transformations on a vector space. It is in the still unchartered realm of nonlinear actions of groups on curved surfaces or manifolds where Professor Zimmer's research will take place.