Connections between Homology Theories for Knots and Three-Manifolds

结和三流形的同调理论之间的联系

• 批准号: 1111680
• 负责人: Stephan Martin Wehrli
• 金额: $ 11.02万
• 依托单位: Syracuse University
• 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: 2011-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1111680&HistoricalAwards=false
• 关键词: Connections; between; Homology; Theories; Knots

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.

Khovanov同源性和Heegaard Floer同源性是针对结和三个manifolds的强大不变性，它们在2000年左右被发现，此后引起了大量的研究活动。特别是，Khovanov同源性和Heegaard Floer同源性都被用来提供有关拓扑构想的新证明，以及在开放的四球上存在异国情调的平滑结构。以前，这种结果仅通过仪表理论可以访问。尽管Khovanov同源性是通过由量子群的代表理论促进的结构来定义的，但Heegaard Floer同源性通过分析方程式解决方案的模量空间进行了分析定义。在2008年左右，Heegaard Floer同源性扩展到具有非空边界的三个manifolds的不变性，称为“边界的漂浮物同源性”，只要将三个manifold分解为三个manifold分解成合适的较小片段，可用于计算Heegaard Floer同源性。该项目的一个主要目标是研究边界的浮点同源性与Khovanov同源性缠结之间的关系。比较这两种理论将更多地了解Khovanov同源性的几何含量，从而使Khovanov同源性更适合应用。此外，边界的浮子同源性与Khovanov同源性之间的设想关系将提供一个可能更一般的联系几何和表示理论之间的联系的例子。该项目的其他目标是开发新的同源理论，用于接触三个manifolds，并分析n台结的Khovanov同源性小组的属性。长期以来，有兴趣分类的拓扑空间，这些拓扑空间是本地三维（例如我们的物理宇宙）或本地四维（例如四维时段）。与对此类空间进行分类的问题有关的问题是将嵌入在给定的三维空间中的打结循环进行分类。在过去的二十年中，数学家使用了来自数学和数学物理学几个不同领域的想法（尤其是从符号几何，量子场理论，弦理论和环量子引力）来开发有力的新工具，以分类结和低维空间。这些新工具中最著名的是Khovanov同源性和Heegaard Floer同源性。该建议旨在研究Khovanov同源性的某些概括与Heegaard Floer同源性之间的联系，并使用这些连接来研究结理论问题。数学结理论在生物医学研究中有应用，在生物医学研究中，它用于研究负责在细胞分裂过程中揭示DNA链的过程。因此，该项目不仅从理论的角度也很重要，而且对于生物医学科学的潜在应用也很重要。