The research program centers around topological questions in algebraic geometry and representation theory. The focus is on the categorical structure of sheaves on singular algebraic varieties. Particular emphasis is placed on loop spaces and other varieties arising in the geometric Langlands program. Several directions of study are considered. The first project applies ideas from the geometric Langlands program to the representation theory of real groups. It develops a geometric form of harmonic analysis for moduli spaces of real bundles on a Riemann surface. The main goal is to further open the representation theory of real groups to the powerful tools of algebraic geometry. The second project seeks to extend the representation theory of loop groups to noncompact groups. Potential applications include the geometric quantization of spaces of flat connections on Riemann surfaces, and the resulting construction of new topological 2+1 dimensional quantum field theories. The third project aims to develop an ``elementary" microlocal theory of perverse sheaves. As a first step in this direction, it seeks an obstruction theory for local systems on the complement of a singular divisor.
Representation theory is the mathematical study of symmetry. Some of the most important phenomena involve places where there is a breakdown of symmetry. For example, many surprising combinatorial formulas result from writing a highly symmetric quantity in terms of contributions from asymmetric pieces. In geometric representation theory, one of the primary tools to measure the breakdown of symmetry is the theory of perverse sheaves. These are complicated objects which detect dynamic aspects of singularities, the locus where symmetry breaks down. The research program described here aims to understand perverse sheaves and their role in representation theory. It seeks to apply perverse sheaves to forms of symmetry which arise in physics. In particular, it aims to develop a geometric version of harmonic analysis and representation theory for symmetries involving loops on a space. The project also seeks elementary ways to think about perverse sheaves from a dynamic perspective.
