CAREER: Heegaard Floer homology and low-dimensional topology

低维拓扑学研究的是三维和四维空间。一个基本目标是了解哪些属性在拉伸或弯曲下保持不变；例如，孔洞的数量是空间的不变属性。这个项目旨在使用一种名为Heegaard Floer同调的代数不变量在拓扑问题上取得进展。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，国际和平研究所计划举办研讨会，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学中占有重要地位。特别是，这一理论对纽结协调的研究产生了影响，导致了关于协调群的丰硕的不变量的构造和结构结果。在最近发展的基础上，PI计划构造新的纽结协调不变量和四球亏格界限。在不同的方向上，PI将研究涉及Heegaard Floer同调和Khovanov同调的谱序列，Khovanov同调是一种组合定义的链不变量。最后，PI旨在为难以计算的结不变量(如解结数字)赋予新的界限。这一奖项反映了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。

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.