Topology Between Dimensions Three and Four

三维和四维之间的拓扑

• 批准号: 2104144
• 负责人: Jennifer Hom
• 金额: $ 38.93万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104144&HistoricalAwards=false
• 关键词: Topology; Between; Dimensions; Three; Four

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.

拓扑学是对形状的研究。低维拓扑是指对 3 维和 4 维空间及其内部的曲线和曲面的研究，其中 3 维和 4 维空间特别令人感兴趣，因为它们足够高，可以实现足够复杂的现象。 （与相对容易理解的维度 1 和维度 2 不同），但又不会高到让有趣的事情变得无趣（维度 5 及以上就是这种情况）。低维拓扑中的基本问题包括：多少次。结必须穿过自身才能解开？关于高维三角测量，三维空间能告诉我们什么？PI 计划研究这些问题，并指导该研究领域的研究生和博士后。会议、讲习班和研讨会，旨在扩大妇女和历史上代表性不足群体成员的参与。PI 计划使用 Heegaard Floer 同源性来研究低维拓扑，以延续已建立的Knot Floer 同调提供了解结界限，并且有界 Floer 同调非常适合研究卫星结。她还建议使用内合 Floer 同调来完善 Manolescu 的反驳。高维三角剖分猜想，给出拓扑流形何时可三角剖分的简单表征。最后，她计划研究。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。