Gravitational Instantons, Mirror Symmetry, and Enumerative Geometry

流形是数学空间；例如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二元性，它将流形上的不同几何领域与其镜像联系在一起。这些几何形状是先验的，彼此无关。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对象。这个项目将主要关注引力瞬子，首先研究它们的微分几何方面，然后通过镜像对称来探索它们在计数几何和代数几何中的含义。此外，该项目还将研究这些含义如何反馈到微分几何中，目的是从几何的不同方面统一对镜像对称的理解。这项研究将为本科生和研究生提供项目。国际数学联合会将继续为本科生组织会议和演示数学的实际应用，以增加下一代几何学家的数量。PI计划使用Syz几何来连接引力瞬子、镜像对称性和计数几何。利用最近在各种引力瞬子中构造的Syz纤颤，PI将研究完全的Syz镜像对称性，同时控制A侧和B侧几何，并与Landau-Ginzburg模型相耦合。PI将用它来理解超Kähler旋转与镜面对称性之间的关系。另一方面，来自镜像对称性的知识将有助于研究对数Calabi-Yau曲面的整体度规描述和引力瞬子的紧致，从而回到引力瞬子的模空间的研究。引力瞬子的度规透视将导致计数几何中局部开放Gromov-Witten不变量的显式计算，以及比较Floer镜族和Gross-Siebert/Gross-Hating-Keel-Siebert镜结构的具体例子。对几何的度量描述的理解的一个副产品是Calabi-Yau流形中极小拉格朗日的各种新构造。该技术的某些部分有望为探索具有纤维结构的Calabi-Yau三重几何结构提供垫脚石。该奖项反映了NSF的法定使命，并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，被认为值得支持。

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.