NSF Award
2204109
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.<br/> <br/>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.<br/><br/>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.
