CAREER: Fukaya Categories and Noncommutative Hodge Structures

在辛几何领域的经典力学中，物理系统的形状和整体行为在很大程度上是由某些被称为伪全纯曲线的面积最小曲面的外观和数量决定的。本研究项目旨在通过建立公式，将不同形状的最小面积曲面（球体、多孔曲面等）的研究简化为更简单类型的最小面积曲面（圆盘）的研究，进一步发展系统的规则，以理解和计数大量物理系统中的此类伪全纯曲线。然后，通过理解后者（磁盘）计数的程度，可以将物理系统分解成基本部分。将研究镜像对称的应用，镜像对称是一种深远的几何对偶性，首先在弦理论中发现，涉及（在一侧）这样的曲线的计数。该项目的教育部分旨在创建一系列关于辛几何和相关领域的在线虚拟研究活动，包括组织虚拟研讨会（正在进行中）、讲习班和迷你课程。PI还将通过传统的讲习班、新课程和研讨会内容、本科和研究生指导以及通过科学展览向K-12学生推广来培训和鼓励数学学生。由该奖项资助的长期研究项目是建立并进一步发展系统框架，用于计算辛几何和镜像对称中的不变量，这些不变量来自伪全纯曲线理论。一方面，它旨在发展和应用新的结构结果，如局部到全局原理，以简化闭辛流形的Fukaya范畴的研究，并将其应用于同调镜像对称。另一方面，该项目旨在进一步阐明Gromov-Witten不变量与Fukaya类别之间的关系，并应用于枚举镜像对称。第三个方向，也是最后一个方向，是研究和进一步解释镜面对称中某些积分格的出现和巧合。许多后一种不变量可以被打包到一个非交换霍奇结构的概念中，为所提出的工作提供了一个有用的框架。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。

The shape and global behavior of physical systems arising in classical mechanics, a purview of the field of symplectic geometry, is known to be largely determined by the appearance and quantity of certain area-minimizing surfaces known as pseudoholomorphic curves. This research project aims to further develop systematic rules for understanding and counting such pseudoholomorphic curves in a large class of physical systems, by establishing formulae reducing the study of area-minimizing surfaces of different shapes (spheres, surfaces with many holes, etc.) to the study of a simpler type of area-minimizing surfaces (disks), and subsequently by understanding the degree to which such latter counts (of disks) can be assembled from a decomposition of the physical system into elementary pieces. Applications will be studied to mirror symmetry, a far-reaching geometric duality first discovered in string theory involving (on one side) counts of such curves. The educational component of the project aims to create a series of online virtual research activities in symplectic geometry and related areas, including organization of a virtual seminar (ongoing), workshops, and mini-courses. The PI will also train and encourage mathematics students through traditional workshops, new course and seminar content, undergraduate and graduate advising, and outreach to K-12 students through science fairs. The long term research project funded by this award is to establish and further develop systematic frameworks for computing invariants in symplectic geometry and mirror symmetry coming from pseudoholomorphic curve theory. In one direction, it aims to develop and apply new structural results such as local-to-global principles to simplify the study of Fukaya categories of closed symplectic manifolds, with applications to homological mirror symmetry. In another, the project aims to further elucidate the relationship between Gromov-Witten invariants and the Fukaya category, with applications to enumerative mirror symmetry. The third and final direction is to study and further explain the appearance and coincidences of certain integral lattices in mirror symmetry. Many of the latter invariants can be packed into the notion of a non-commutative Hodge structure, giving a useful framework for the proposed work.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.