Quantum topological structures in geometric representation theory

当前的研究是在超对称规范理论的启发下探索几何表示理论的新方向。它还提出了对拉格朗日膜的深谷分类的新方法，并对镜像对称产生了影响。具体研究包括基于特征束的特征变异的傅立叶分析，椭圆曲线上环群经束的特征束理论，拉格朗日交理论的新的局部和同局部模型。这些方法主要是代数和拓扑的，但受到谐波分析和微局部分析中发现的基本模式的启发。潜在的应用范围从椭圆曲线上束的特征变异的上同调和范畴量化的朗兰兹对偶到不依赖全纯盘的深谷范畴的束理论重表述。本研究旨在促进数学与物理之间的互动，并教育学生使用同调几何的新工具。它的重点包括在规范理论，谐波分析和朗兰兹程序的十字路口的对象。活动包括进一步阐述量子场论等重要但困难的主题，以及为不同领域的学生提供与知名研究人员互动的机会。很难估计数学和物理以外的影响，但这项研究与计算拓扑及其通过小样本理解大数据集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.