Heegaard Floer homology and its applications to low-dimensional topology

本课题研究低维拓扑中的不变量，这些不变量来自规范理论和辛几何，特别是Heegaard flower同调。重点是Heegaard flower同调在低维拓扑中的应用，以及Heegaard flower同调与低维拓扑其他方面的联系。我们计划研究的一个问题是heegard花同调的不平衡缝合流形分解。这与使用Heegaard flower同源性表征不可压缩曲面有关。我们将研究的另一个问题是Heegaard flower同源性中的植物学问题，即在多大程度上可以利用Heegaard flower同源性来确定流形。这些问题与Dehn手术和Khovanov同源性有关。我们还将讨论Floer同调在四维拓扑中的应用，例如K3表面上结手术的拓扑。在微观世界中，大分子通常被视为三维空间中的结和链接。因此，本项目研究的结不变量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是，大分子的联锁有多牢固，如何检测它们的手性，以及如何改变它们的拓扑结构。这些特征在纳米技术和药理学领域具有极其重要的意义。

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.