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Floer Homology and Immersed Curve Invariants in Low Dimensional Topology

低维拓扑中的Floer同调和浸没曲线不变量

基本信息

批准号：
2105501
负责人：
Jonathan Hanselman
金额：
$ 20.03万
依托单位：
Princeton University
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：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105501&HistoricalAwards=false
关键词：
Floer Homology Immersed Curve Invariants

项目摘要

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.

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打结曲线进行分类的工具。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。这些三维对象的研究，通过分析某些不变的属性，捕捉对象的基本特征。一个成功的家庭这样的不变量，这导致了许多应用，因为它是第一次开发二十年前，被称为Heegaard Floer同源。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。该项目旨在重新制定这些不变量中的一些，既揭示了它们的底层结构，也提供了计算工具，并导致新的应用。特别地，PI的目标是将边界Heegaard Floer不变量转化为具有边界的3-流形，其形式为复杂的代数对象，这个项目的重点是Heegaard Floer同调的结构，目的是扩展和更好地理解拓扑量子场论（TQFT）-Heegaard Floer理论中的相似结构，并将其应用于低维拓扑问题。首先，PI旨在将早期的结果扩展到Heegaard Floer同调的更强的“负”版本，它携带更多的信息，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，PI将探索在卫星操作和和谐下的结的不变量的行为。一个长期的目标是研究曲面和它们之间的态射的对称积的福谷范畴，找到这些物体的实际几何描述，这些描述适合于Heegaard Floer同调的描述，如（2+1+1）-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进行评估，被认为值得支持审查标准。