Homological Invariants of Knots and Three-Manifolds

DMS-0603940 Jacob Rasmussen这个项目是关于一类纽结不变量，称为“纽结同调”，它推广了经典的不变量，如Alexander和Jones多项式。PI计划调查这些不变量的两个版本之间的猜想关系。第一个版本是由Khovanov和Rozansky开发的，本质上是组合的。第二种称为纽结Floer同调，起源于Ozsvath和Szabo的Heegaard Floer同调。它不是组合的，但众所周知，它携带着大量关于结的几何信息。尽管这两种理论的定义不同，但它们确实显示出一些惊人的相似之处。该项目旨在解释这些相似之处，并利用它们来更好地理解这两种理论。一个可能的应用是寻找规范理论不变量的组合类似，如纽结Floer同调。三维空间中的纽结曲线的研究与三维和四维空间本身的几何密切相关。在过去的几十年里，来自物理学的想法导致了这类曲线的两种主要类型的不变量的发展，即所谓的“量子”不变量和“规范理论”不变量。尽管这两种类型都植根于量子场论，但几乎没有迹象表明它们之间存在关联。然而，最近，两者之间的一些显著相似之处已经开始显现。这个项目的目的是更好地理解这两类不变量之间的关系。

DMS-0603940Jacob RasmussenThis project is about a class of knot invariants known as ``knot homologies'' which generalize classical invariants such as theAlexander and Jones polynomial. The PI plans to investigate a conjectured relationship between two versions of these invariants. The first version was developed by Khovanov and Rozansky and is combinatorial in nature. The second is known as knot Floer homology, and has its origins in the Heegaard Floer homology of Ozsvath and Szabo. It is not combinatorial, but is known to carry a great deal of geometric information about the knot. Despite the differences in their definitions, these two theories exhibit some truly striking similarities. The project aims to explain these similarities and to use them to get a better understanding of both theories. One posssible application is to find combinatorial analogs of gauge theoretic invariants like the knot Floer homology.The study of knotted curves in three-dimensional space is intimately related to the geometry of three- and four-dimensional spaces themselves. In the last few decades, ideas from physics have led to the development of two major types of invariants for such curves, known as ``quantum'' and ``gauge theoretic'' invariants. Although both types have roots in the quantum theory of fields, there was little sign that they were related. Recently, however, some remarkable similarities between the two have begun to appear. The aim of this project is to better understand the relationship between these two classes of invariants.