The Structure of the Gromov-Witten Invariants

被称为辛流形的物体在现代几何和物理学的研究中是非常重要的。该奖项支持辛几何和弦理论的交叉研究，辛几何为经典力学提供了框架，弦理论作为统一广义相对论和粒子物理的潜在候选者发展起来，使研究时空和相空间模型成为可能，而这些模型是其他方法无法获得的。所考虑的问题是深层次的和基础性的，对这些问题的回答将导致不同数学领域之间的新技术和相互作用，并将在弦理论中有潜在的应用。这项工作有许多有趣的应用，继续这一研究方向的许多有希望的方向都涉及到学生。首席研究员的研究和外联活动将对下一代数学家的教育产生影响，使他们在毕业生涯早期从事前沿研究。本研究的主题涉及辛流形和与其相关的不变量的结构。具体地说，其中一个项目旨在了解Gromov-Witten不变量的多种类型的一般类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的，并激发了人们对数学的浓厚兴趣，以及理解它们背后的几何学的许多尝试。第二个项目旨在了解辛几何中Gromov-Witten虚拟基本循环的性质，并探索其函数论方面。它包括一套建议的公理，这套公理将唯一地决定它，灵感来自于同调理论的艾伦伯格-斯蒂罗德公理。封闭辛流形的Gromov-Witten虚拟基本类有许多不同的构造，每一种都有自己的优势和优势，因此了解这些构造中的大多数承载相同的信息是有用的，允许人们选择最适合特定问题的构造。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。

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 are deep and of foundational nature, and answers to these will lead to new techniques and interactions among different fields of mathematics and will have potential applications in string theory. There are many interesting applications of this work, and many of the promising directions for continuing this line of research, involve 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 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. A second 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.