Connections between Homology Theories for Knots and Three-Manifolds

Khovanov同源性和Heegaard flower同源性是结和三流形的强大不变量，它们在2000年左右被发现，并激起了大量的研究活动。特别是，利用Khovanov同调和Heegaard Floer同调给出了拓扑Milnor猜想和开放四球上奇异光滑结构存在的新证明。以前，这样的结果只能通过规范理论得到。Khovanov同调是通过量子群表示理论的构造组合定义的，而Heegaard flower同调是通过微分方程解的模空间解析定义的。2008年前后，Heegaard Floer同调被推广到具有非空边界的三流形的不变量，称为“有边Floer同调”，它可以用于组合计算Heegaard Floer同调，只要给定一个三流形分解成合适的小块。本课题的主要目的是研究缠结的有边Floer同调和Khovanov同调之间的关系。通过对这两种理论的比较，可以更好地揭示Khovanov同调的几何内容，从而使Khovanov同调更适合于应用。此外，设想的边界Floer同调和Khovanov同调之间的关系将为辛几何和表示理论之间可能更普遍的联系提供一个例子。本项目的其他目标是发展新的接触三流形的同调理论，并分析n-结索的Khovanov同调群的性质。数学家长期以来一直对局部三维（如我们的物理宇宙）或局部四维（如四维时空）的拓扑空间分类感兴趣。与这种空间的分类问题相关的是对嵌入在给定三维空间中的结环进行分类的问题。在过去的二十年里，数学家们使用了来自数学和数学物理的几个不同领域的思想（特别是辛几何、量子场论、弦理论和环量子引力）来开发强大的新工具来分类结和低维空间。这些新工具中最引人注目的是Khovanov同源性和Heegaard flower同源性。本文旨在探讨Khovanov同调的某些推广与Heegaard flower同调之间的联系，并利用这些联系研究结理论问题。数学结理论在生物医学研究中有应用，在那里它被用来研究细胞分裂过程中负责解开DNA链的过程。因此，该项目不仅从理论角度来看很重要，而且对其在生物医学科学中的潜在应用也很重要。

Khovanov homology and Heegaard Floer homology are powerful invariants for knots and three-manifolds, which were discovered around the year 2000, and which have since stirred a tremendous amount of research activity. In particular, both Khovanov homology and Heegaard Floer homology have been used to give new proofs of the topological Milnor conjecture and of the existence of exotic smooth structures on the open four-ball. Previously, such results had only been accessible via gauge theory. While Khovanov homology is defined combinatorially, via a construction which is motivated by the representation theory of quantum groups, Heegaard Floer homology is defined analytically, through moduli spaces of solutions of differential equations. Around 2008, Heegaard Floer homology was extended to an invariant for three-manifolds with non-empty boundary, called "bordered Floer homology", which can be used to compute Heegaard Floer homology combinatorially whenever a decomposition of a three-manifold into suitable smaller pieces is given. A main goal of this project is to study the relationship between bordered Floer homology and Khovanov homology for tangles. Comparing these two theories will expectedly shed more light on the geometric content of Khovanov homology and thus make Khovanov homology more suited to applications. Moreover, the envisioned relationship between bordered Floer homology and Khovanov homology will provide an example of a perhaps more general connection between symplectic geometry and representation theory. Other goals of this project are to develop new homology theories for contact three-manifolds, and to analyze the properties of Khovanov homology groups of n-cables of knots.Mathematicians have long been interested in classifying topological spaces that are locally three-dimensional (like our physical universe) or locally four-dimensional (like four-dimensional space-time). Related to the problem of classifying such spaces is the problem of classifying knotted loops embedded in a given three-dimensional space. Over the past two decades, mathematicians have used ideas coming from several different areas of mathematics and mathematical physics (in particular from symplectic geometry, quantum field theory, string theory, and loop quantum gravity) to develop powerful new tools for classifying knots and low-dimensional spaces. The most notable ones among these new tools are Khovanov homology and Heegaard Floer homology. This proposal aims to investigate the connections between certain generalizations of Khovanov homology and Heegaard Floer homology, and to use these connections to study knot theoretical problems. Mathematical knot theory has applications in biomedical research, where it is used to study the processes that are responsible for unraveling DNA strands during cell division. Thus, this project is important not only from a theoretical perspective, but also for its potential applications to biomedical sciences.