Perverse Sheaves in Representation Theory

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The research program centers around topological questions in algebraicgeometry and representation theory. The focus is on the categoricalstructure of sheaves on singular algebraic varieties. Particular emphasisis placed on loop spaces and other varieties arising in the geometricLanglands program. Several directions of study are considered.The first project applies ideas from the geometric Langlands program tothe representation theory of real groups. It develops a geometric form ofharmonic analysis for moduli spaces of real bundles on a Riemann surface.The main goal is to further open the representation theory of real groupsto the powerful tools of algebraic geometry. The second project seeks toextend the representation theory of loop groups to noncompact groups.Potential applications include the geometric quantization of spaces offlat connections on Riemann surfaces, and the resulting construction ofnew topological 2+1 dimensional quantum field theories. The thirdproject aims to develop an ``elementary" microlocal theory of perversesheaves. As a first step in this direction, it seeks an obstruction theoryfor local systems on the complement of a singular divisor.Representation theory is the mathematical study of symmetry. Some of themost important phenomena involve places where there is a breakdown ofsymmetry. For example, many surprising combinatorial formulas result fromwriting a highly symmetric quantity in terms of contributions fromasymmetric pieces. In geometric representation theory, one of the primarytools to measure the breakdown of symmetry is the theory of perversesheaves. These are complicated objects which detect dynamic aspects ofsingularities, the locus where symmetry breaks down. The research programdescribed here aims to understand perverse sheaves and their role inrepresentation theory. It seeks to apply perverse sheaves to forms ofsymmetry which arise in physics. In particular, it aims to develop ageometric version of harmonic analysis and representation theory forsymmetries involving loops on a space. The project also seeks elementaryways to think about perverse sheaves from a dynamic perspective.