Structural Results in Floer Theory and Mirror Symmetry

辛几何是一个广泛关注物理系统的全局形状的领域，例如那些跟踪约束配置中粒子的可能位置和动量的领域。从数学物理学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，困难的微分方程的出现使这些工具具有挑战性。 该项目旨在通过开发系统的计算规则（例如，剪切和粘贴），并通过建立不同类型的弗洛尔理论之间的新关系。 利用这些，该项目旨在解决有关弗洛尔理论结构的开放问题，并找到镜像对称（首次出现在弦理论中的显着几何对偶）和辛几何中奇点（突变）研究的新应用。PI还将通过研讨会、新课程和研讨会内容以及K-12科学博览会的评审来培训和鼓励数学学生。该项目旨在利用辛几何中各种形式奇点研究的输入（及其应用），在Floer理论和镜像对称中开发新的结构结果。 在一个方向上，该项目的目的是表明，包装福谷类别满足预期的范-坎彭风格的局部性，并推导出作为一个结果的公理和层理论的特点福谷类别的斯坦流形（部分使用其拉格朗日的奇异结构）。 在一个相关的方向上，该项目旨在几何计算与某些Landau-Ginzburg模型相关的福谷类别的Hochschild不变量（即，全纯函数），并推导出新的应用程序的研究奇异性等功能。 第三个也是最后一个方向是理解隐藏奇点（在有一个奇异的镜子的意义上）对福谷类别的影响，以及计算和结构上的后果。这个奖项反映了NSF的法定使命，并被认为值得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来支持。

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.