Perverse Sheaves in Representation Theory

研究项目围绕代数几何和表示论中的拓扑问题。重点是奇异代数簇上层的范畴结构。特别强调几何朗兰兹纲领中出现的循环空间和其他变体。几个方向的研究被认为是。第一个项目适用的想法从几何朗兰兹计划的代表性理论的真实的群体。它给出了黎曼曲面上真实的丛的模空间的调和分析的几何形式，其主要目的是进一步将真实的群的表示理论引入代数几何的有力工具中。第二个项目是将圈群的表示理论推广到非紧群，潜在的应用包括黎曼曲面上的平面连通空间的几何量子化，以及由此产生的新的拓扑2+1维量子场论的构造。第三个项目的目标是发展一个“初级”的反常层微局部理论。作为这一方向的第一步，它寻求一个障碍theoryfor局部系统上的一个奇异因子的补集。一些最重要的现象涉及到对称性破坏的地方。例如，许多令人惊讶的组合公式都是由一个高度对称的量的对称部分的贡献而产生的。在几何表示论中，测量对称性破坏的主要工具之一是反常层理论。这些复杂的物体可以探测到奇异点的动态方面，对称性被破坏的轨迹。这里描述的研究计划旨在理解反常层及其在表示理论中的作用。它试图将反常层应用于物理学中出现的对称形式。特别是，它的目的是发展几何版本的调和分析和表示理论的对称性涉及循环的空间。该项目还寻求从动态角度思考反常层的基本方法。

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.