Floer Homology and Immersed Curve Invariants in Low Dimensional Topology

这个低维拓扑学的研究项目开发了用于对三维空间以及其中的曲面和结点曲线进行分类的工具。这些拓扑物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了解聚合物和DNA的打结。这些三维物体是通过分析某些捕捉物体本质特征的不变性质来研究的。这类不变量的一个成功家族称为Heegaard Floer同调，自二十年前首次发展以来，它已经导致了大量的应用。Heegaard Floer同调的定义借鉴了多个领域的复杂技术，包括拓扑学、几何学和分析。这个项目旨在以一种既揭示其底层结构又提供计算工具并导致新应用的方式来重新表述其中的一些不变量。具体地说，PI的目的是将具有边界的三维流形的有边界Heegaard Floer不变量转化为由曲面上的曲线集合构造的几何对象。本项目主要研究Heegaard Floer同调的结构，目的是扩展和更好地理解Heegaard Floer理论中的类拓扑量子场论(TQFT)结构，并将其应用于低维拓扑问题。首先，PI的目的是将早期的结果推广到Heegaard Floer同调的更强的“负”版本，它承载了更多的信息，也与四维不变量有有趣的联系。PI还将考虑具有较高亏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节点的不变量的行为。一个长期的目标是研究曲面的对称乘积的Fukaya类别和它们之间的态射，找到这些对象的实用几何描述，这些对象符合作为(2+1+1)维拓扑量子场论的Heegaard Floer同调的描述。该奖项反映了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。

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.