Topology Between Dimensions Three and Four

拓扑学是对形状的研究。低维拓扑是指研究三维和四维空间，以及其中的曲线和曲面。维度3和维度4是特别有趣的，因为它们足够高，可以允许足够复杂的现象（不像维度1和维度2，它们相对容易理解），但又没有高到让有趣的事情变得无趣（这是维度5及以上的情况）。低维拓扑学的基本问题包括：一个结要穿过自己多少次才能解开？关于高维三角形，三维空间能告诉我们什么？PI计划调查这些问题，并在这个研究领域指导研究生和博士后。她还将组织会议、讲习班和研讨会，以扩大妇女和历史上代表性不足的群体成员的参与。PI计划使用Heegaard flower同源性来研究低维拓扑，作为既定计划的延续。结花同源性提供了解结界限，而镶边花同源性非常适合研究卫星结。PI将使用这些工具来调查解开卫星结的数量。她还建议使用对合花同调来改进Manolescu对高维三角剖分猜想的反驳，给出了拓扑流形何时可三角剖分的简单表征。最后，她计划研究积分和有理结协调与同源配合群的区别。该奖项反映了美国国家科学基金会的法定使命，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。

Topology is the study of shapes. Low-dimensional topology refers to the study of 3- and 4- dimensional spaces, and curves and surfaces inside of them. Dimensions 3 and 4 are of particular interest because they are high enough to allow sufficiently complex phenomena (unlike dimensions 1 and 2, which are relatively well understood), yet not so high that interesting things become uninteresting (which is the case in dimensions 5 and above). Fundamental questions in low-dimensional topology include: How many times must a knot pass through itself in order to become untangled? What can 3-dimensional spaces tell us about higher dimensional triangulations? The PI plans to investigate such questions, and to mentor graduate students and postdocs in this field of study. She will also organize conferences, workshops, and seminars, with an aim to broadening the participation of women and members of historically underrepresented groups.The PI plans to use Heegaard Floer homology to study low-dimensional topology, in continuation of an established program. Knot Floer homology provides unknotting bounds, and bordered Floer homology is well suited for studying satellite knots. The PI will use these tools in tandem to investigate the unknotting number of satellite knots. She also proposes to use involutive Floer homology to refine Manolescu’s disproof of the high dimensional triangulation conjecture, giving a simple characterization of when a topological manifold is triangulable. Lastly, she plans to study the differences between the integral and rational knot concordance and homology cobordism groups.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.