Perverse Sheaves in Representation Theory

研究计划围绕代数几何和表示论中的拓扑问题展开。重点研究了奇异代数簇上的层的范畴结构。特别强调在几何语言程序中出现的循环空间和其他变量。考虑了几个研究方向：第一个项目将几何朗兰兹程序的思想应用到实群的表示理论中。给出了黎曼曲面上实丛的模空间的调和分析的几何形式，其主要目的是将实群的表示理论进一步推向代数几何的强大工具。第二个项目旨在将循环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。第三个项目旨在发展变态重病的“初级”微局部理论。作为这一方向的第一步，它在奇异因子的补集上寻找局部系统的障碍理论。表示理论是对对称性的数学研究。一些最重要的现象涉及到对称性破裂的地方。例如，许多令人惊讶的组合公式的结果是写出了一个高度对称的量，就是不对称部分的贡献。在几何表示理论中，衡量对称性破缺的主要工具之一是倒立理论。这些都是复杂的物体，可以探测奇点的动态方面，奇点是对称性破坏的轨迹。这里所描述的研究计划旨在了解倒置排列及其在表示理论中的作用。它试图将倒置的薄片应用于物理学中出现的对称形式。特别是，它的目的是为空间上涉及环的对称性发展调和分析和表示理论的年龄测量版本。该项目还寻求从动态的角度思考弯曲滑轮的基本方法。

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.