CAREER: Fukaya Categories and Noncommutative Hodge Structures

在经典力学中出现的物理系统的形状和全局行为，辛几何领域的一个范围，已知在很大程度上取决于某些面积最小化表面的外观和数量，称为伪全纯曲线。该研究项目旨在通过建立公式来减少对不同形状（球体、具有许多孔的表面等）的面积最小化表面的研究，进一步开发用于理解和计算一大类物理系统中的这种伪全纯曲线的系统规则。到一个更简单的类型的面积最小化的表面（磁盘）的研究，并随后通过理解的程度，这种后者的计数（磁盘）可以组装从物理系统分解成基本件。将研究镜像对称的应用，这是一种影响深远的几何对偶性，首先在弦理论中发现，涉及（一侧）这种曲线的计数。该项目的教育部分旨在创建一系列辛几何及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）、讲习班和迷你课程。PI还将通过传统研讨会，新课程和研讨会内容，本科生和研究生咨询以及通过科学博览会向K-12学生推广来培训和鼓励数学学生。该奖项资助的长期研究项目是建立和进一步发展用于计算辛几何和镜像对称中来自伪全纯曲线理论的不变量的系统框架。在一个方向上，它的目的是开发和应用新的结构结果，如局部到整体的原则，以简化研究的福谷类别的封闭辛流形，应用同调镜像对称。另一方面，该项目旨在进一步阐明Gromov-Witten不变量和福谷范畴之间的关系，并应用于枚举镜像对称。第三个也是最后一个方向是研究和进一步解释镜像对称中某些积分晶格的出现和重合。许多后者的不变量可以打包成一个非交换霍奇结构的概念，为拟议的工作提供了一个有用的框架。这个奖项反映了NSF的法定使命，并已被认为是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的支持。

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.