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，同时保持某些内在性质。通过添加时间维度，很自然地将变形视为在一端具有一个对象，在另一端具有另一个形状的四维对象。低维拓扑学中的一个基本问题是，我们是否可以建立一个连接两个给定三维对象的四维空间。Heegaard Floer同调是研究这类问题的重要工具。它给了拓扑学家一种知道两个三维空间不能被一个四维空间联系起来的方法。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刻的联系。本项目旨在开发计算Heegaard Floer同调的新工具。这个项目的一个主要重点是开发一个新的减版本的边界Heegaard Floer同源的三维空间与环面边界组件。这个理论是基于马诺列斯库和奥兹瓦的链接手术公式。该项目旨在利用该理论研究Némethi的格同调猜想。此外，该项目的目的是研究对称性在这个理论和链接手术公式。有了这些工具，PI希望给出可计算的同调配边不变量。此外，PI致力于指导研究生和本科生，以及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。

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.