The Structure of the Gromov-Witten Invariants

Gromov-Witten 不变量的结构

基本信息

批准号：
1905361
负责人：
Eleny-Nicoleta Ionel
金额：
$ 38万
依托单位：
Stanford University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2024-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1905361&HistoricalAwards=false
关键词：
Structure Gromov Witten Invariants

项目摘要

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.

在现代几何和物理学的研究中，称为符号流形的对象至关重要。该奖项支持在符号几何形状的交集的研究，该几何形状为经典力学提供了框架和弦理论的框架，该理论发展为统一一般相对论和粒子物理学的潜在候选者，从而使其他方式无法通过其他方式来学习时空和相位空间的模型。所考虑的问题是深层的和基本的性质，对这些问题的答案将导致不同数学领域之间的新技术和相互作用，并将在字符串理论中具有潜在的应用。这项工作有许多有趣的应用程序，以及许多继续进行这一研究的有前途的方向，都涉及学生。首席研究者的研究和外展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尖端研究。这项研究的主题涉及符合歧管和与之相关的不变性的结构。具体而言，其中一个项目旨在了解Gromov-Witten不变性的许多口味的一般类型结构定理背后的几何原因。这些类型的结构是使用弦理论猜想的，并引起了人们对数学的浓厚兴趣，并尝试理解其背后的几何形状。第二个项目旨在了解格罗莫夫（Gromov-witten）虚拟基本周期在符号几何形状中的特性，并探索其功能方面。它包括一组提议的公理，这些公理会独特地确定它，灵感来自同源理论的Eilenberg-Steenrod公理。 Gromov-Witten的虚拟基本封闭形式歧管有许多不同的结构，每个封闭式歧管都具有其自己的优势和优势，因此知道这些结构中的大多数具有相同的信息都是有用的，因此，人们可以选择最适合特定问题的任何构造，这表明NSF的法定任务和审查的范围都在评估范围内，这是通过评估的范围来进行的。