Floer Homology and Immersed Curve Invariants in Low Dimensional Topology

这个低维拓扑的研究项目开发了用于分类三维空间以及其中的表面和打结曲线的工具。这些拓扑对象的应用范围从探索我们生活的三维空间宇宙的可能形状到理解聚合物和DNA的打结。通过分析捕捉物体本质特征的某些不变属性来研究这些三维物体。这种不变量的一个成功的家族被称为Heegaard flower同源，自20年前首次开发以来，它已经导致了许多应用。Heegaard flower同源的定义借鉴了来自多个领域的复杂技术，包括拓扑、几何和分析。该项目旨在以一种既能揭示其底层结构又能提供计算工具并导致新应用的方式重新表述其中的一些不变量。特别是，PI旨在将具有边界的3-流形的有边界Heegaard flower不变量（采用复杂代数对象的形式）转换为由曲面上曲线集合构建的几何对象。本项目主要研究Heegaard flower同调结构，旨在扩展和更好地理解Heegaard flower理论中的拓扑量子场论（TQFT）类结构，并将其应用于低维拓扑问题。首先，PI的目标是将早期的结果扩展到Heegaard flower同调的更强的“负”版本，它携带了更多的信息，并且与四维不变量也有有趣的联系。PI还将考虑具有高格边界或具有多个边界分量的流形的情况。在一个相关的方向上，PI将探索在卫星操作和协调下结的不变量的行为。一个长期的目标是研究曲面的对称积和它们之间的态射的Fukaya类别，找到这些对象的实用几何描述，这些描述适合Heegaard flower同调的描述作为（2+1+1）维拓扑量子场论。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。

This research project in low-dimensional topology develops tools for classifying three-dimensional spaces as well as surfaces and knotted curves within them. Applications of these topological objects range from exploring the possible shapes of the three-dimensional spatial universe we live in to understanding the knotting of polymers and DNA. These three-dimensional objects are studied by analyzing certain invariant properties which capture essential characteristics of the objects. A successful family of such invariants, which has lead to numerous applications since it was first developed two decades ago, is called Heegaard Floer homology. The definition of Heegaard Floer homology draws on sophisticated techniques from multiple fields, including topology, geometry, and analysis. This project aims to reformulate some of these invariants in a way that both sheds light on their underlying structure and also provides computational tools and leads to new applications. In particular, the PI aims to translate bordered Heegaard Floer invariants for 3-manifolds with boundary, which take the form of complicated algebraic objects, into geometric objects built from collections of curves in a surface.This project focuses on the structure of Heegaard Floer homology with the goal to extend and better understand the Topological Quantum Field Theory (TQFT)-like structure in Heegaard Floer theory and to apply it to problems in low-dimensional topology. First, the PI aims to extend the earlier results to the stronger “minus” version of Heegaard Floer homology, which carries more information and also has interesting connections to four-dimensional invariants. The PI will also consider the case of manifolds with higher genus boundary or with multiple boundary components. In a related direction, the PI will explore the behavior of invariants for knots under the satellite operation and concordance. A long-term goal is to study the Fukaya categories of symmetric products of surfaces and morphisms between them, finding practical geometric descriptions of these objects which fit into a description of Heegaard Floer homology as a (2+1+1)-dimensional Topological Quantum Field Theory.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.