Mathematical Sciences: Geometry and Representations of Reductive Groups

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Abstract Evens 9623322 Evens will continue his work on the relation between geometry and representation theory. In his work with Grojnowski, he will study the relationship with multiplicities of restrictions of representations of real semisimple groups and dimensions of cohomology groups of certain spaces. In his work with Mirkovic, he will study characteristic cycles for nilpotent orbits and their relation to certain problems in representation theory. He will also study analogous problems for representations of Hecke algebras. 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.