Real Gromov-Witten Theory and its Applications

弦理论是一种通过振动弦来表示基本粒子的模型，其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，但它还没有做出实验上可检验的预测。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。这个项目的两个方向将进一步测试弦理论的数学，旨在验证其预测的全纯曲线在各种设置的行为，并调查不同数学领域之间的预期联系，包括枚举代数几何，纽结理论和表示理论。该奖项还将支持PI的研究生。这个项目将建立在PI以前的工作基础上，这些工作建立了弦理论真实的部分背后的数学基础，以推进对这个部分产生的所谓镜像对称预测的数学理解。它还将利用PI的十年之久的工作，建立了BCOV镜像对称预测的计数亏格1曲线在著名的五次三倍，并继续最近完成的工作的拓扑德利涅-具有共轭标记点对的稳定真实的曲线的Mumford模空间。该奖项反映了NSF的法定使命，并通过使用基金会的知识产权进行评估，被认为值得支持。优点和更广泛的影响审查标准。

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’s two directions will further test string theory mathematically, aiming to verify its predictions for the behavior of holomorphic curves in various settings and to investigate the expected connections between different fields of mathematics, including enumerative algebraic geometry, knot theory, and representation theory. This award will also support the PI's graduate students. This project will build on the PI’s previous work, which established the mathematical foundations behind the real sector of string theory, to advance the mathematical understanding of the so-called mirror symmetry predictions arising from this sector. It will also utilize the PI’s decade-old work, which established the BCOV mirror symmetry prediction for counts of genus 1 curves in the renown quintic threefold and continue the recently completed work on the topology of Deligne-Mumford moduli spaces of stable real curves with conjugate pairs of marked points.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.