CAREER: Heegaard Floer homology and low-dimensional topology

拓扑学是研究不同空间的形状。一维空间和二维空间以及五维以上的空间都是很好理解的;粗略地说，在前者中，没有足够的维度来产生有趣的现象，而在后者中，有如此多的维度，以至于任何有趣的东西都有足够的空间变得无趣。低维拓扑学主要关注三维和四维，在这些空间中会出现许多独特的现象。一个中心问题是，当人们允许第四维时，三维中的打结环是否会变得不打结。你也可以问，在空间中剪出一个结，然后以不同的方式填充由此产生的空隙时会发生什么。纽结理论适用于非常小的（例如，DNA的打结链的行为）以及极大的（例如，宇宙的形状）。在与研究组成部分，PI计划进一步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，并通过领导当地的初中和高中学生的数学活动。她还将为本科生组织一个研讨会，介绍他们的研究，并了解更多关于数学职业的信息。由Ozsvath和Szabo开发的Heegaard Floer同源性是理解低维拓扑的强大工具。PI计划使用几个最新的发展来回答该领域长期存在的问题。例如，最近定义的亨德里克斯和马诺列斯库的对合Heegaard Floer同调可以应用于理解和谐群中的整除性。在另一个方向，PI计划使用Manolescu和Ozsvath的链接手术公式来研究哪些流形来自n分量链接上的手术。她还建议研究同源配边，这是密切相关的结一致性，希望提供障碍，同源配边手术的结。

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.