Moduli Spaces of Holomorphic Curves: Properties and Applications

弦理论是一种模型，它通过振动弦来表示基本粒子，目的是统一自然界的四种基本力。虽然弦理论是当今物理学的主要范式之一，但它尚未做出可实验验证的预测。然而，它产生了许多数学预测，导致了代数几何和辛拓扑的基本发展，特别是在（伪）全纯曲线方面。该项目旨在进一步从数学上测试弦理论，同时深化对这些曲线的数学理解，着眼于将其应用于更经典的几何问题。这项工作中的一些项目将由研究生和其他初级研究人员与研究者合作进行。本课题在代数几何、辛拓扑和弦理论的结合处有四个不同的方向。它将探索辛拓扑中伪全纯曲线的刚性与两族代数几何之间的联系。本文将研究2及以上属的稳定态射的模空间的局部结构，目的是为了以后在镜像对称和计数几何中的应用。PI还将把他的计算格罗莫夫-威滕不变量的方法应用到更多的目标上，目的是在其他情况下验证弦理论的预测。第四个方向是发展正属实Gromov-Witten理论及其与开放弦理论和实枚举几何的关系。该奖项由代数与数论和几何分析项目共同资助。

String theory is a model that represents elementary particles by vibrating strings with the aim of unifying the four fundamental forces of nature. While string theory is one of the main paradigms in physics today, it has yet to make experimentally testable predictions. However, it has generated many mathematical predictions that have led to fundamental developments in algebraic geometry and symplectic topology, especially in relation to (pseudo-) holomorphic curves. This project aims to further test string theory mathematically, while deepening the mathematical understanding of such curves with an eye toward applications to more classical problems in geometry. Some of the projects in this work will be pursued by graduate students and other junior researchers in collaboration with the investigator.This project has four distinct directions at the juncture of algebraic geometry, symplectic topology, and string theory. It will explore connections between the rigidity of pseudo-holomorphic curves in symplectic topology and birational algebraic geometry. It will study the local structure of moduli spaces of stable morphisms of genus 2 and higher, with the aim of later applications in mirror symmetry and enumerative geometry. The PI will also apply his method for computing genus 1 Gromov-Witten invariants to more targets, with the aims of verifying predictions of string theory predictions in additional cases. The fourth direction aims to develop positive-genus real Gromov-Witten theory and its relations with open string theory and real enumerative geometry.This award is jointly funded by the Algebra and Number Theory and Geometric Analysis programs.