Moduli Spaces of Pseudoholomorphic Maps

被称为辛流形的数学对象在现代几何和物理的研究中具有根本的重要性。该奖项支持辛几何和弦理论交叉领域的研究，辛几何为经典力学和弦理论提供了框架，弦理论是统一广义相对论和粒子物理学的潜在候选者，使时空和相空间模型的研究能够通过其他方式实现。在这个项目中考虑的问题是基础性的，这些问题的答案有望导致新的技术和不同数学领域之间的相互作用，并在弦理论中有潜在的应用。这项工作有几个有趣的应用，以及继续这条研究路线的有希望的方向，其中一些涉及研究生。首席研究员的研究和推广活动将对下一代数学家的教育产生影响，使他们在研究生生涯的早期参与前沿研究。本研究的主题涉及辛流形及其不变量的结构。具体来说，其中一个项目旨在了解许多Gromov-Witten不变量的一般Gopakumar-Vafa型结构定理背后的几何原因。这些类型的结构是用弦理论推测出来的，激发了人们对数学的兴趣，并试图理解它们背后的几何结构。另一个项目旨在了解辛几何中Gromov-Witten虚拟基本循环的性质，并探索其功能方面。它包括一组被提议的公理，这些公理将唯一地决定它，灵感来自于同调理论的Eilenberg-Steenrod公理。Gromov-Witten闭辛流形的虚拟基本类有许多不同的构造，每一个都有自己的长处和优点，所以知道大多数这些构造携带相同的信息是有用的，允许人们选择最适合特定问题的任何构造。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical objects known as symplectic manifolds are of fundamental importance in the study of modern geometry and physics. This award supports research that lies at the intersection of symplectic geometry, which provides the framework for classical mechanics, and string theory, which developed as a potential candidate for unifying general relativity and particle physics, enabling the study of models of space-time and phase-spaces inaccessible by other means. The problems considered in this project are of foundational nature, and answers to these are expected to lead to new techniques and interactions among different fields of mathematics and have potential applications in string theory. There are several interesting applications of this work, and promising directions for continuing this line of research, some of them involving graduate students. The research and outreach activities of the principal investigator will have an impact on the education of next generation of mathematicians, engaging them in cutting-edge research early in their graduate career.The theme of this research involves symplectic manifolds and the structure of the invariants associated to them. Specifically, one of the projects aims to understand the geometric reasons behind a general Gopakumar-Vafa type structure theorem for many flavors of Gromov-Witten invariants. These types of structures were conjectured using string theory and have spurred a lot of interest in mathematics and many attempts to understand the geometry behind them. Another project aims to understand the properties of the Gromov-Witten virtual fundamental cycle in symplectic geometry and explore its functorial aspects. It includes a proposed set of axioms that would uniquely determine it, inspired by the Eilenberg-Steenrod axioms for homology theories. There are many different constructions of the Gromov-Witten virtual fundamental class of closed symplectic manifolds, each one with its own strengths and advantages, so knowing that most of these constructions carry the same information is useful, allowing one to choose whatever construction may be best suited for the particular problem.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.