Mathematical Sciences: Topological Methods in RepresentationTheory and Automorphic Forms

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The main goal of this project is to develop methods from topology and apply them to representation theory of groups and automorphic forms. The first part is a program to study perverse sheaves from the micro-local point of view. This includes an extension of the category of perverse sheaves to arbitrary homogeneous symplectic manifolds as well as the study of the mixed and irregular versions of the theory. The second part attempts applications to representation theory, in particular, geometric constructions of representations and micro-local study of characters of representations. The third part is work on the geometric Langlands conjecture from various points of view. Groups are the natural algebraic structures to describe symmetry. It is a classical method of understanding a group to realize (or represent) it as a collection of transformations of a vector space. The theory of representations has become highly evolved and even made itself indispensable in the study of quantum mechanics, but it is by no means a completed theory and a closed book. The investigator is bringing to bear some of the latest topological and geometric methods in answering outstanding questions of representation theory. Connections with other topics in algebra and analysis are to be expected.