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Structural Results in Floer Theory and Mirror Symmetry

弗洛尔理论和镜像对称的结构结果

基本信息

批准号：
1907635
负责人：
Sheel Ganatra
金额：
$ 22.8万
依托单位：
University of Southern California
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907635&HistoricalAwards=false
关键词：
Structural Results Floer Theory Mirror

项目摘要

Symplectic geometry is an area broadly concerned with the global shape of physical systems, such as those tracking the possible positions and momenta of particles in a constrained configuration. Insights from mathematical physics have led to powerful tools, collectively called Floer theory, for extracting properties of such systems (for instance the number of periodic orbits associated to a system with given kinetic and potential energy); unfortunately the appearance of difficult differential equations makes these tools challenging to apply. This project seeks to simplify the study of Floer theory of a large class of spaces by developing systematic rules for computation (via e.g., cut and paste) and by establishing new relationships between different types of Floer theory. Using these, the project aims to solve open problems about the structure of Floer theory and find new applications to mirror symmetry (a remarkable geometric duality first arising in string theory) and the study of singularities (abrupt changes) in symplectic geometry. The PI will also train and encourage mathematics students through workshops, new course and seminar content, and judging of K-12 science fairs.This project aims to develop new structural results in Floer theory and mirror symmetry using input from (and with applications to) the study of singularities of various forms in symplectic geometry. In one direction, the project aims to show that wrapped Fukaya categories satisfy expected van-Kampen style locality properties, and deduce as a consequence axiomatic and sheaf-theoretic characterizions of Fukaya categories of Stein manifolds (using in part the singular structure of their Lagrangian skeleta). In a related direction, the project aims to geometrically calculate the Hochschild invariants of Fukaya categories associated to certain Landau-Ginzburg models (i.e., holomorphic functions), and deduce new applications to the study of singularities of such functions. The third and final direction is to understand the effect of hidden singularities (in the sense of having a singular mirror) on Fukaya categories, with computational and structural consequences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

辛几何是一个与物理系统的全局形状广泛相关的领域，例如那些跟踪受约束构型中粒子的可能位置和动量的物理系统。来自数学物理的见解导致了强大的工具，统称为Floer理论，用于提取此类系统的属性(例如，与给定动能和势能的系统相关的周期轨道的数量)；不幸的是，困难的微分方程式的出现使这些工具的应用具有挑战性。该项目旨在通过制定系统的计算规则(例如，通过剪切和粘贴)以及通过在不同类型的Floer理论之间建立新的关系来简化对一大类空间的Floer理论的研究。利用这些，该项目旨在解决关于Floer理论结构的公开问题，并找到镜像对称性(一种最早出现在弦理论中的显著的几何对偶)和研究辛几何中的奇点(突变)的新应用。PI还将通过工作坊、新的课程和研讨会内容以及K-12科学博览会的评判来培训和鼓励数学学生。这个项目的目的是利用辛几何中各种形式的奇点研究的投入和应用来开发Floer理论和镜像对称的新结构结果。在一个方向上，该项目的目的是证明包裹的Fukaya范畴满足预期的van-Kampen型局部性性质，并由此推出Stein流形的Fukaya范畴的公理和层论特征(部分利用其拉格朗日骨架的奇异结构)。在一个相关的方向上，该项目旨在几何地计算与某些Landau-Ginzburg模型(即全纯函数)相关的Fukaya范畴的Hochschild不变量，并在研究这类函数的奇异性方面得出新的应用。第三个也是最后一个方向是了解隐藏的奇点(在有一面奇异的镜子的意义上)对Fukaya类别的影响，以及计算和结构上的结果。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。