Three-Dimensional Manifolds, Heegaard Floer Homology and Knot Theory

三维流形、Heegaard Floer 同调和纽结理论

• 批准号: 1904628
• 负责人: Zoltan Szabo
• 金额: $ 30.8万
• 依托单位: Princeton 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: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904628&HistoricalAwards=false
• 关键词: Three; Dimensional; Manifolds; Heegaard; Floer

Some of the central objects studied in the project are knotted circles in the three-dimensional space. These mathematical objects are used to study more complicated three- and four-dimensional spaces called manifolds that form the foundations of an area of mathematics known as low dimensional topology. Building on the principal investigator's prior work with Peter Ozsvath, this project will use modern gauge-theoretical and topological methods to study various open problems and conjectures related to these spaces, investigate recent methods and developments, and develop new invariants for knots and links. The project also aims to enhance graduate student research by introducing them to different areas of low dimensional topology and teaching recent methods in knot and link Floer homology and Heegaard Floer theory. Heegaard Floer homology and knot Floer homology constructions use methods from Topology and Symplectic Geometry through the study of Heegaard diagrams and holomorphic disks in symmetric products of a Heegaard surface. There have been some recent breakthroughs in knot Floer homology, and the principal investigator will apply these new algebraic and symplectic techniques to study problems in Low Dimensional Topology. Specifically, the PI plans to generalize bordered Floer homology methods to links, with potential applications to the Thurston norm and other topological questions; develop tools to compute the master complex for knot Floer homology using the Pong algebra; study the double point knot Floer homology developed by Lipschitz; and use the HF mixed invariant to study exotic smooth structures on 4-manifolds. Other projects include investigating the Berge conjecture, cosmetic surgery conjecture, and mutations.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.

项目中研究的一些中心物体在三维空间中是打结的圆圈。这些数学对象被用来研究更复杂的三维和四维空间，称为流形，它们构成了被称为低维拓扑的数学领域的基础。在首席研究员Peter Ozsvath之前的工作基础上，该项目将使用现代测量理论和拓扑方法来研究与这些空间相关的各种开放问题和猜想，调查最近的方法和发展，并为结和链接开发新的不变量。该项目还旨在加强研究生的研究，向他们介绍低维拓扑的不同领域，并教授结和链接花同调和Heegaard花理论的最新方法。通过研究Heegaard曲面的对称积中的Heegaard图和全纯盘，利用拓扑学和辛几何中的方法构造Heegaard花同调和结花同调。最近在结花同调方面取得了一些突破，首席研究员将应用这些新的代数和辛技术来研究低维拓扑问题。具体来说，PI计划将有边界的flower同调方法推广到链路，并有可能应用于Thurston范数和其他拓扑问题；开发了利用Pong代数计算结花同调主复形的工具；研究Lipschitz提出的双点结花同源性；利用HF混合不变量研究了4流形上的奇异光滑结构。其他项目包括调查贝尔热猜想、整容手术猜想和突变。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。

项目成果

Purely cosmetic surgeries and pretzel knots

纯粹的整容手术和椒盐卷饼结

• DOI: 10.2140/pjm.2021.313.195
• 发表时间: 2021
• 期刊: Pacific Journal of Mathematics
• 影响因子: 0.6
• 作者: Stipsicz, András I.;Szabó, Zoltán
• 通讯作者: Szabó, Zoltán