Gauge Theory, Floer Homology, and Invariants of Low-Dimensional Manifolds

规范理论、Floer 同调和低维流形不变量

• 批准号: 1949209
• 负责人: Jianfeng Lin
• 金额: $ 3.69万
• 依托单位: University of California-San Diego
• 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 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1949209&HistoricalAwards=false
• 关键词: Gauge; Theory; Floer; Homology; Invariants

This NSF award funds research to study three- and four-dimensional spaces with knotted curves and surfaces embedded within them. Such objects are of particular interest to scientists because of their connection to our physical world and the space-time continuum. These studies will deepen the understanding of our universe. Moreover, the theory of knotted curves has seen broad applications in other fields of science, for example, in investigations of the structure of DNA and that of protein molecules. In recent years, several new techniques have been introduced to study these low dimensional spaces. This project will be devoted to further developing the techniques, providing surprising insights and proving unexpected relationships between existing theories. In collaborative research, the PI will explore new invariants that would help us distinguish between different spaces.This project is devoted to enhancing the power of gauge theory and Floer homology and applying these tools to the study of three- and four-dimensional manifolds. In the first part of this project, joint with Tirasan Khandhawit and Hirofumi Sasahira, the PI will further develop the theory of unfolded Seiberg-Witten Floer spectrum invariants for general three-manifolds and use it to draw new conclusions regarding four-manifolds. In the second part of this project, joint with Daniel Ruberman and Nikolai Saveliev, the PI will carry out research on invariants of four-manifolds with homology identical to the space obtained as a product of a circle with a three sphere, with surprising applications in the study of the three-dimensional homology cobordism group. In the third part of this project, the PI will seek for a sheaf theoretic description of invariants from gauge theory, with the goal to make these invariants more flexible and powerful.

该NSF奖项基金研究研究了三维和四维空间，其中嵌入了打结的曲线和表面。由于科学家与我们的物理世界和时空连续体的联系，因此这些物体特别感兴趣。这些研究将加深对我们宇宙的理解。此外，打结的曲线理论在其他科学领域中看到了广泛的应用，例如，在DNA的结构和蛋白质分子的结构中。近年来，已经引入了几种新技术来研究这些低维空间。该项目将致力于进一步开发这些技术，提供令人惊讶的见解并证明现有理论之间的意外关系。在协作研究中，PI将探索新的不变式，以帮助我们区分不同的空间。该项目致力于增强量规理论和浮动同源性的力量，并将这些工具应用于三维和四维流形的研究。在该项目的第一部分中，与Tirasan Khandhawit和Hirofumi Sasahira的联合，PI将进一步发展为一般的三个manifolds的展开的seiberg-witten浮子光谱不变性的理论，并使用它来提出有关四个manifolds的新结论。 在该项目的第二部分中，与丹尼尔·鲁贝曼（Daniel Ruberman）和尼古拉·萨维耶夫（Nikolai Saveliev）的联合，PI将对四个manifolds的不变式进行研究，其同源性与带有三个球体的圆圈的乘积相同，在对三维同型共同群的研究中具有令人惊讶的应用。在该项目的第三部分中，PI将寻求对轨距理论的不变理论描述，目的是使这些不变性更加灵活和强大。