Quantum topological structures in geometric representation theory

目前的研究寻求新的方向的几何表示理论的启发超对称规范理论。它还提出了新的方法，福谷类别的拉格朗日膜的镜像对称的后果。具体的研究包括傅立叶分析的字符品种的字符层，理论的字符层循环组通过捆绑椭圆曲线，新的本地和同伦模型拉格朗日相交理论。该方法主要是代数和拓扑，但灵感来自谐波分析和微局部分析中发现的基本模式。潜在的应用范围从Langlands对偶的上同调的字符品种和范畴量化的椭圆曲线上的一个层理论的重新制定的福谷范畴没有上诉holomorphic disks.The研究旨在进一步之间的相互作用数学和物理和教育学生的同伦几何的新工具。它的重点包括对象在十字路口的规范理论，谐波分析，和朗兰兹计划。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既定研究人员互动的机会。很难估计数学和物理学之外的影响，但这项研究与计算拓扑学及其在通过小样本理解大数据集方面的应用有潜在的联系。

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.