CAREER: Fukaya Categories and Noncommutative Hodge Structures

经典力学中出现的物理系统的形状和整体行为是辛几何领域的一个范畴，众所周知，它在很大程度上由某些被称为伪全纯曲线的面积最小化曲面的出现和数量决定。本研究项目旨在通过建立公式，减少对不同形状(球体、多孔曲面等)的面积最小化曲面的研究，进一步发展系统的规则，以便在一大类物理系统中理解和计算这种伪全纯曲线。涉及到更简单类型的面积最小化表面(盘)的研究，并随后通过理解这种(盘的)计数可以从物理系统的分解成基本碎片来组装的程度来进行。将研究镜像对称性的应用，这是一种影响深远的几何对偶，最先在弦理论中发现，涉及(一侧)此类曲线的计数。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，包括组织一次虚拟研讨会(正在进行)、讲习班和小型课程。PI还将通过传统的讲习班、新的课程和研讨会内容、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系统的框架，用于计算来自伪全纯曲线理论的辛几何和镜像对称中的不变量。在一个方向上，它的目的是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。在另一个方面，该项目旨在进一步阐明Gromov-Witten不变量与Fukaya范畴之间的关系，并将其应用于枚举镜对称。第三个也是最后一个方向是研究和进一步解释某些积分晶格在镜像对称下的出现和重合。这一奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.