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特别令人感兴趣，因为它们的高度足以允许足够复杂的现象(与维度1和维度2不同，后者相对容易理解)，但又不会太高，以至于有趣的事情变得不有趣(维度5和更高的情况就是这种情况)。低维拓扑中的基本问题包括：一个纽结必须穿过自身多少次才能解开？关于高维三角剖分，3维空间能告诉我们什么？PI计划调查这些问题，并指导这一研究领域的研究生和博士后。她还将组织会议、研讨会和研讨会，旨在扩大妇女和历史上代表性不足群体的成员的参与。PI计划继续现有的计划，使用Heegaard Floer同源来研究低维拓扑。纽带Floer同调提供了解开边界，而有边带Floer同调非常适合于研究卫星结。PI将同时使用这些工具来调查卫星结的解开数量。她还建议利用对合Floer同调来改进Manolescu对高维三角剖分猜想的反证，给出了拓扑流形何时是三角剖分的简单刻画。最后，她计划研究整体和有理纽结调和以及同调协和群之间的差异。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。

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.