Geometry and Representations of Reductive Groups

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AbstractEvensEvens will continue his work on the relation between geometry and representation theory. In his work with Lu, he will attempt to generalize their previous theorem relating the new field of Poisson geometry to the classical 19th century problem of computing intersection multiplicities. In particular, he hopes to find a Poisson structure to help compute quantum cohomology of the flag manifold. In his work with Grojnowski, he will continue his study of multiplicities of restrictions of representations of real semisimple groups, with the goal of solving certain multiplicity one problems. He will study characteristic cycles for nilpotent orbits and their relation to certain problems in representation theory.The use of symmetry in the study of geometrical problems has been one of the great unifying themes of modern mathematics. Ideas based on considerations of symmetry have had a profound impact on elementary questions in arithmetic and on physics, especially to quantum mechanics. There are certain topological constructions which are analogous to the problem of counting points in a finite set. The investigator plans to apply these constructions to problems motivated by number theory and theoretical physics.