Gravitational Instantons, Mirror Symmetry, and Enumerative Geometry

流形是数学空间;例子包括我们的三维世界和四维时空。镜像对称是弦理论中的一种神秘的对偶性，它将流形上的不同几何领域与其镜像联系起来。这些几何形状是彼此先验无关的。引力瞬子是一种特殊的四维流形，是理论物理中量子引力理论的基石，也是数学不同分支中的重要研究对象。这个项目将主要关注引力瞬子，从研究它们的微分几何方面开始，然后通过镜像对称探索枚举几何和代数几何的含义。此外，该项目将研究这些含义如何反馈到微分几何，旨在统一几何学不同方面对镜像对称性的理解。这项研究将为本科生和研究生提供项目。PI将继续组织会议，并说明数学的实际应用本科生，以增加下一代geometers池。PI计划使用SYZ几何来连接引力瞬子、镜像对称和枚举几何。利用最近在各种引力瞬子中已经构建的SYZ纤维化，PI将研究一个完整的SYZ镜像对称，同时控制A面和B面几何，并与Landau-Ginzburg模型耦合。PI将使用它来理解超凯勒旋转与镜像对称之间的关系。另一方面，镜像对称性的知识将有助于研究log Calabi-Yau曲面的整体度规描述和引力瞬子的紧化，进而研究引力瞬子的模空间。引力瞬子的度规观点将导致在枚举几何中显式计算局部开放Gromov-Witten不变量，并给出比较Floer镜和Gross-Siebert/Gross-Hacking-Keel-Siebert镜结构的具体例子。理解几何的度量描述的副产品是各种新的最小拉格朗日在卡-丘流形的建设。该技术的某些部分预计将提供踏脚石探测卡-丘三重几何与纤维结构。这一奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Manifolds are mathematical spaces; examples include our three-dimensional world and the four-dimensional space-time. Mirror symmetry is a mysterious duality from string theory that links different fields of geometry on a manifold and its mirror image. These geometries are a priori unrelated to each other. Gravitational instantons are special four-dimensional manifolds that are the building blocks of quantum gravity theory in theoretical physics and important objects in different branches of mathematics. This project will mainly focus on gravitational instantons, starting by investigating their differential geometric aspects and then probing implications in enumerative geometry and algebraic geometry via mirror symmetry. Furthermore, the project will study how these implications feed back to differential geometry, aiming to unify the understanding of mirror symmetry from different aspects in geometry. The research will provide projects for undergraduate and graduate students. The PI will continue to organize conferences and to illustrate practical applications of mathematics for undergraduate students to increase the pool of next generation geometers. The PI plans to use the SYZ geometry to bridge the connection between gravitational instantons, mirror symmetry, and enumerative geometry. Utilizing the SYZ fibrations already constructed in various gravitational instantons recently, the PI will investigate a full SYZ mirror symmetry, simultaneously governing both A-side and B-side geometry and coupled with the Landau-Ginzburg model. The PI will use it to understand the relation between hyper-Kähler rotation with mirror symmetry. On the other hand, the knowledge from the mirror symmetry will help to study the global metric description of log Calabi-Yau surfaces and the compactification of gravitational instantons, in return to the study of the moduli space of gravitational instantons. The metric perspective of the gravitational instantons will lead to explicit calculation of local open Gromov-Witten invariants in enumerative geometry and concrete examples for comparing the family Floer mirrors and Gross-Siebert/Gross-Hacking-Keel-Siebert mirror constructions. A byproduct of the understanding of the metric description of the geometry is various new constructions of minimal Lagrangians in Calabi-Yau manifolds. Some parts of the techniques are expected to provide steppingstones for probing the Calabi-Yau three-fold geometry with fibration structures.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.