CAREER: Heegaard Floer homology and low-dimensional topology

拓扑学是对不同空间形状的研究。一维和二维空间是很好理解的，五维及以上的空间也是如此。粗略地说，在前者中，没有足够的维度来容纳有趣的现象，而在后者中，维度太多，任何有趣的事物都有足够的空间变得无趣。低维拓扑侧重于三维和四维，其中会发生许多独特的现象。一个核心问题是，当允许存在第四维时，三维中的打结环是否会变得不打结。人们还可以问，在空间中切出一个结，然后以不同的方式填充由此产生的空隙，会发生什么。结理论既适用于非常小的事物（例如，DNA 打结链的行为），也适用于非常大的事物（例如，宇宙的形状）。在开展研究工作的同时，PI 计划进一步加强指导和推广工作，例如，监督本科生和研究生的研究，以及领导当地中学生和高中生的数学活动。她还将为本科生组织一个研讨会，展示他们的研究成果并了解更多有关数学职业的信息。Heegaard Floer 同调由 Ozsvath 和 Szabo 开发，是理解低维拓扑的强大工具。 PI 计划利用最近的几项进展来回答该领域长期存在的问题。例如，最近定义的 Hendricks 和 Manolescu 的对合 Heegaard Floer 同调可能适用于理解索引群中的整除性。在另一个方向上，PI 计划使用 Manolescu 和 Ozsvath 的连接手术公式来研究 n 分量连接上的手术会产生哪些流形。她还提出研究与绳结一致性密切相关的同源协调性，希望为绳结手术的同源协调性提供障碍。

Topology is the study of the shape of different spaces. One- and two-dimensional spaces are well-understood, as are dimensions five and above; roughly, in the former, there are not enough dimensions for interesting phenomena, and in the latter, there are so many dimensions that anything interesting has enough room to become uninteresting. Low-dimensional topology focuses on three- and four-dimensions, where many unique phenomena occur. One central question is whether a knotted loop in three-dimensions becomes unknotted when one allows a fourth dimension. One can also ask what happens upon cutting a knot out of space, and then filling in the resulting void in a different way. Knot theory has applications to the very small (e.g., the behavior of knotted strands of DNA) as well as the extremely large (e.g., the shape of the universe). In tandem with the research component, the PI plans to further her mentoring and outreach efforts, for example, by supervising undergraduate and graduate research, and by leading local math events for middle and high school students. She will also organize a workshop for undergraduates to present their research and learn more about careers in mathematics.Heegaard Floer homology, developed by Ozsvath and Szabo, is a powerful tool for understanding low-dimensional topology. The PI plans to use several recent developments to provide answers to long-standing questions in the field. For example, the recently defined involutive Heegaard Floer homology of Hendricks and Manolescu may have applications to understanding divisibility in the concordance group. In a different direction, the PI plans to study which manifolds arise from surgery on an n-component link, using the link surgery formula of Manolescu and Ozsvath. She also proposes to study homology cobordism, which is closely related to knot concordance, with the hope of providing obstructions to being homology cobordant to surgery on a knot.