New Perspectives in Heegaard Floer Homology

拓扑学中的一个基本问题是我们是否可以将一种形状变形为另一种形状，同时保留某些内在属性。通过添加时间维度，我们很自然地将变形视为一个 4 维对象，其一端有一个对象，另一端有另一个形状。低维拓扑中的一个基本问题是我们是否可以构建一个连接两个给定的 3 维对象的 4 维空间。 Heegaard Floer 同源性是研究此类问题的重要工具。它为拓扑学家提供了一种方法来了解两个 3 维空间不能通过 4 维空间关联。 Heegaard Floer 同调涉及微分方程复杂解的计数，与 Seiberg-Witten 理论、Yang-Mills 理论和数学物理有着深厚的联系。该项目旨在开发计算 Heegaard Floer 同调的新工具。该项目的主要重点是为具有环面边界分量的 3 维空间开发新的有边界 Heegaard Floer 同调的负版本。该理论基于Manolescu 和Ozsváth 的链接手术公式。该项目旨在利用该理论研究Némethi的晶格同调猜想。此外，该项目旨在研究该理论和链接手术公式中的对称性。通过这些工具，PI 希望给出可计算的同源共边不变量。此外，PI 还致力于指导研究生和本科生，并组织活动来传播知识。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。

A fundamental question in topology is whether we can deform one shape into another while preserving certain intrinsic properties. By adding a time dimension, it is natural to think of the deformation as being a 4-dimensional object which has one object at one end, and the other shape at the other end. A fundamental question in low-dimensional topology is whether we can build a 4-dimensional space which connects two given 3-dimensional objects. Heegaard Floer homology is an important tool for studying such questions. It gives topologists a way of knowing that two 3-dimensional spaces cannot be related by a 4-dimensional space. Heegaard Floer homology involves the counts of complicated solutions to differential equations and has deep connections to Seiberg-Witten theory, Yang-Mills theory, and mathematical physics.This project aims to develop new tools for computing Heegaard Floer homology. A main focus of this project is to develop a new minus version of bordered Heegaard Floer homology for 3-dimensional spaces with torus boundary components. This theory is based on the link surgery formula of Manolescu and Ozsváth. The project aims to use this theory to study the lattice homology conjecture of Némethi. Additionally, the project aims to study symmetries in this theory and in the link surgery formula. With these tools, the PI hopes to give computable invariants of homology cobordism. In addition, the PI endeavors to mentor graduate students and undergraduates, as well as organize events to disperse knowledge.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.