Structural Results in Floer Theory and Mirror Symmetry

辛几何是一个与物理系统的整体形状广泛相关的领域，例如那些在受限构型中跟踪粒子的可能位置和动量的领域。来自数学物理的见解导致了强大的工具，统称为弗洛尔理论，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，复杂微分方程的出现使得这些工具难以应用。该项目旨在通过开发系统的计算规则（例如，剪切和粘贴）以及建立不同类型的花理论之间的新关系，简化对大空间类的花理论的研究。利用这些，该项目旨在解决关于Floer理论结构的开放问题，并在镜像对称（弦理论中首先出现的一种显著的几何对偶性）和辛几何奇点（突变）的研究中找到新的应用。PI还将通过研习班、新课程和研讨会内容、K-12科学博览会评审等方式培养和鼓励数学学生。该项目旨在利用辛几何中各种形式的奇点研究的输入（并应用于），在flower理论和镜像对称中开发新的结构结果。在一个方向上，该项目旨在证明包裹的Fukaya类别满足预期的van-Kampen风格局部性，并推断出Stein流形的Fukaya类别的公理和束理论特征（部分使用其拉格朗日骨架的奇异结构）。在一个相关的方向上，该项目旨在几何计算与某些Landau-Ginzburg模型（即全纯函数）相关的Fukaya类别的Hochschild不变量，并推导出这些函数的奇异性研究的新应用。第三个也是最后一个方向是理解隐藏的奇点（在有一个奇异镜像的意义上）对Fukaya类别的影响，以及计算和结构的后果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.