CAREER: Heegaard Floer homology and low-dimensional topology

职业：Heegaard Florer 同调和低维拓扑

• 批准号: 2237131
• 负责人: Linh Truong
• 金额: $ 55万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237131&HistoricalAwards=false
• 关键词: CAREER; Heegaard; Floer; homology; low

Low-dimensional topology is the study of spaces of dimensions three and four. A fundamental goal is to understand what properties remain unchanged under stretching or bending; for example, the number of holes is an invariant property of a space. This project aims to make progress on topological questions using an algebraic invariant called Heegaard Floer homology. In parallel with its research goals, the project includes mentoring and outreach activities. In addition, the PI plans to organize workshops which will provide speaking and collaboration opportunities for junior researchers. Heegaard Floer homology plays a prominent role in modern low-dimensional topology. In particular, the theory has been influential in the study of knot concordance, leading to the construction of fruitful invariants and structural results on the concordance group. Building on recent developments, the PI plans to construct new knot concordance invariants and four-ball genus bounds. In a different direction, the PI will study spectral sequences involving Heegaard Floer homology and Khovanov homology, a combinatorially-defined link invariant. Finally, the PI aims to give new bounds on difficult-to-compute knot invariants such as unknotting number.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.

低维拓扑学是对三维和四维空间的研究。一个基本的目标是理解在拉伸或弯曲下什么性质保持不变;例如，孔的数量是空间的不变性质。这个项目的目的是取得进展的拓扑问题使用代数不变称为Heegaard弗洛尔同源。在实现研究目标的同时，该项目还包括辅导和外联活动。此外，PI计划组织研讨会，为初级研究人员提供演讲和合作机会。Heegaard Floer同调在现代低维拓扑中起着重要的作用。特别是，该理论在纽结和谐的研究中产生了影响，导致了和谐群上富有成效的不变量和结构结果的构建。基于最近的发展，PI计划构建新的结一致不变量和四球属界。在不同的方向，PI将研究涉及Heegaard Floer同调和Khovanov同调的谱序列，这是一种组合定义的链接不变量。最后，PI旨在为难以计算的结不变量（如解结数）提供新的界限。该奖项反映了NSF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。