Quantum topological structures in geometric representation theory

几何表示论中的量子拓扑结构

基本信息

批准号：
1319287
负责人：
David Nadler
金额：
$ 13.06万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2012
资助国家：
美国
起止时间：
2012-07-01 至 2015-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319287&HistoricalAwards=false
关键词：
Quantum topological structures geometric representation

项目摘要

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.

当前的研究追求了受超对称仪理论启发的几何表示理论的新方向。它还提出了福卡亚类别拉格朗日麸皮类别的新方法，对镜子对称性产生了后果。具体的研究包括对角色系束系的角色品种的傅立叶分析，通过椭圆曲线上的束束循环组的性格束系理论以及拉格朗日交叉路口理论的新的局部和同型模型。这些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的基本模式。潜在的应用范围从Langlands二重性涉及角色品种的共同体学和椭圆曲线上的捆绑量的分类量化，再到福卡亚类别的捆绑理论重新重新制定，而无需吸引Holomorphic disk。该研究的目的是在数学和物理学之间进一步互动，并在新工具中进行新工具的互动。它的重点包括对量规理论，谐波分析和兰兰兹计划的十字路口的对象。活动包括进一步讲述重要但困难的主题，例如量子场理论，以及不同领域的学生与已建立研究人员互动的机会。很难估计数学和物理学之外的影响，但是该研究具有与计算拓扑结构及其应用的潜在联系，以通过小样本了解大型数据集。