Quantum topological structures in geometric representation theory

目前的研究受到超对称规范理论的启发，为几何表示理论寻求新的方向。它还提出了拉格朗日膜的Fukaya范畴的新方法，其结果是镜像对称性。具体研究内容包括特征标集的傅里叶分析，椭圆曲线上通过丛的循环群的特征标集理论，以及拉格朗日交集理论的新的局部和同伦模型。这些方法主要是代数和拓扑学的，但灵感来自于调和分析和微局部分析中的基本模式。潜在的应用范围从特征标簇的上同调和椭圆曲线上丛的范畴量化的朗兰兹对偶，到不求助于全纯圆盘的Fukaya范畴的层论重构。本研究旨在进一步促进数学和物理之间的相互作用，并教育学生使用新的同伦几何工具。它的重点包括处于规范理论、调和分析和朗兰兹计划十字路口的天体。活动包括进一步阐述重要但困难的主题，如量子场论，以及为不同领域的学生提供与知名研究人员互动的机会。很难在数学和物理之外估计影响，但这项研究与计算拓扑学及其通过小样本理解大数据集的应用有潜在联系。

The current research pursues new directions for geometric representation theory inspired by supersymmetric gauge theory. It also proposes new approaches to Fukaya categories of Lagrangian branes with consequences for mirror symmetry. Specific research includes a Fourier analysis of character varieties in terms of character sheaves, a theory of character sheaves for loop groups via bundles on elliptic curves, and new local and homotopical models for Lagrangian intersection theory. The methods are primarily algebraic and topological, but inspired by basic patterns found in harmonic analysis and microlocal analysis. Potential applications range from Langlands dualities for the cohomology of character varieties and categorical quantizations of bundles on elliptic curves to a sheaf-theoretic reformulation of Fukaya categories without appeal to holomorphic disks.The research aims to further interactions between mathematics and physics and to educate students in the new tools of homotopical geometry. Its focus includes objects at the crossroads of gauge theory, harmonic analysis, and the Langlands program. Activities include further exposition of important but difficult topics such as quantum field theory, as well as opportunities for students in diverse areas to interact with established researchers. It is difficult to estimate the impact outside of mathematics and physics, but the research has potential links to computational topology and its applications to understanding large data sets through small samples.