Representation theory via topological field theory

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。拟议的研究计划解决了代数几何和表示理论中的拓扑问题。重点研究规范理论中出现的模空间上的层范畴的结构。本课程的目的是将拓扑场论和导出的代数几何应用于实李群和复李群的表示理论。该方法利用几何朗兰兹程序的两个互补观点，利用其丰富的结构作为一种四维拓扑场理论和作为调和分析的几何类比。另一个目的是建立在辛几何和奇点理论之间的新兴词典之上，获得对Fukaya范畴和可构造层的结构的新见解。应用包括可积系统的量子化，以及建立基本的倒置理论的可能性。所提出的程序有助于快速扩展经典几何的量子类比的研究。它的灵感来自于朗兰兹程序，该程序将对称空间的调和分析与数域的伽罗瓦对称性联系起来，并从镜像对称的数学物理及其高维前件中得到启发。包括研究对称性和量子系统的数学家在内的许多数学家都对游戏中的技术感兴趣。该程序还提供了与从事弦和D-膜工作的高能物理学家的许多接口点。拟议研究的主要更广泛的影响是教育，重点是为学生在相关领域工作做好准备。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposed research program addresses topological questions in algebraic geometry and representation theory. The focus is on the structure of categories of sheaves on moduli spaces arising in gauge theory. The goal is to apply topological field theory and derived algebraic geometry to the representation theory of real and complex Lie groups. The approach exploits two complementary viewpoints on the Geometric Langlands Program, taking advantage of its rich structure as a 4d topological field theory and as a geometric analogue of harmonic analysis. Another aim is to build upon the emerging dictionary between symplectic geometry and singularity theory, obtaining new insights into the structure of the Fukaya category and constructible sheaves. Applications include quantization of integrable systems, and the possibility of an elementary theory of perverse sheaves.The proposed program contributes to the rapidly expanding study of quantum analogues of classical geometry. It draws inspiration from both the Langlands program relating the harmonic analysis of symmetric spaces with the Galois symmetries of number fields and the mathematical physics of mirror symmetry and its higher dimensional antecedents. The techniques in play are of interest to a broad range of mathematicians including those that study symmetries and quantum systems. The program also offers many points of interface with high energy physicists working on strings and D-branes. The primary broader impact of the proposed research is educational, with an emphasis on the preparation of students to work in related areas.