CAREER: Heegaard Floer homology and low-dimensional topology

拓扑学研究的是不同空间的形状。一维和二维空间都很好理解，五维及以上空间也是如此；粗略地说，在前者中，没有足够的维度来呈现有趣的现象，而在后者中，有太多的维度，以至于任何有趣的东西都有足够的空间变得无趣。低维拓扑主要关注三维和四维，在那里会发生许多独特的现象。一个核心问题是，当一个人允许进入第四维空间时，三维空间中打结的环路是否会被解开。你也可以问，在从空间中剪出一个结，然后以不同的方式填充由此产生的空隙时，会发生什么。结理论可以应用于非常小的（例如，DNA链打结的行为）和非常大的（例如，宇宙的形状）。与研究部分相结合，PI计划进一步加强她的指导和推广工作，例如，通过监督本科生和研究生的研究，以及为初中和高中学生领导当地的数学活动。她还将为本科生组织一个研讨会，展示他们的研究成果，并了解更多关于数学职业的知识。由Ozsvath和Szabo提出的Heegaard flower同调是理解低维拓扑的有力工具。PI计划利用最近的几项发展为该领域长期存在的问题提供答案。例如，最近定义的Hendricks和Manolescu的对合Heegaard flower同调可以应用于理解一致性群中的可除性。在另一个方向上，PI计划使用Manolescu和Ozsvath的连杆手术公式，研究n分量连杆上的手术产生了哪些流形。她还建议研究与结一致性密切相关的同源配合，希望为结手术的同源配合提供障碍。

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.