Heegaard Floer homology and its applications to low-dimensional topology

Heegaard Florer 同调及其在低维拓扑中的应用

基本信息

批准号：
1103976
负责人：
Yi Ni
金额：
$ 21.47万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2011
资助国家：
美国
起止时间：
2011-07-01 至 2015-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103976&HistoricalAwards=false
关键词：
Heegaard Floer homology its applications

项目摘要

This project deals with invariants in low-dimensional topology which come from gauge theory and symplectic geometry, especially Heegaard Floer homology. The focus will be the applications of Heegaard Floer homology to low-dimensional topology, and the connection between Heegaard Floer homology and other aspects of low-dimensional topology. One problem we plan to study is unbalanced sutured manifold decompositions for Heegaard Floer homology. This is related to characterizing incompressible surfaces using Heegaard Floer homology. Another problem we will study is the botany problem in Heegaard Floer homology, namely, to what extent one may determine a manifold using its Heegaard Floer homology. These problems are related to questions about Dehn surgeries and Khovanov homology. We will also address the applications of Floer homology to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.In the microscopic world, macromolecules are often visualized as knots and links in the three-dimensional space. The knot invariants studied in this project thus provide important tools in analyzing the structures of macromolecules: Some questions our methods can study are, how firmly the macromolecules are interlocked, how to detect their chirality, and how to change their topological structure. These features are extremely significant in nanotechnology and pharmacology.

该项目涉及低维拓扑的不变式，这些拓扑来自量规理论和符号几何形状，尤其是Heegaard浮动的同源性。重点将是Heegaard Floer同源性在低维拓扑中的应用，以及Heegaard Floer同源性与低维拓扑的其他方面之间的联系。我们计划研究的一个问题是Heegaard Floer同源性的缝合流形分解不平衡。这与使用Heegaard Floer同源性表征不可压缩的表面有关。我们将研究的另一个问题是Heegaard Floer同源性中的植物学问题，即，使用其Heegaard Floer同源性可以在多大程度上决定多种多样。这些问题与有关Dehn手术和Khovanov同源性的问题有关。我们还将解决浮点同源性在4维拓扑的应用，例如，在K3表面上的结手术拓扑。在微观世界中，大分子通常被视为结和三维空间中的链接。因此，在该项目中研究的结式不变剂为分析大分子的结构提供了重要的工具：我们的方法可以研究的一些问题，大分子互锁的牢固，如何检测其手性以及如何改变其拓扑结构。这些特征在纳米技术和药理学中非常重要。