Noncommutative Low-Dimensional Topology

低维拓扑学的一个基本目标是流形的分类。流形的基本群是一个重要的代数不变量，但很少是阿贝尔群。另一方面，由于流形的同调群是交换群，所以它们很容易分类，但在考虑同调群时，基本群的许多丰富结构被丢失了。具有局部系数的同调将模与流形联系起来。由于模是阿贝尔群，区分它们仍然是成立的。此外，由于这些通常是非交换环上的模，基本群的一些丰富结构被保留下来。这些同调模，以及在它们上面定义的连接形式，已经被用来定义纽结补、3-流形和代数曲线补的代数不变量。这些都导致了新的结果结和谐，障碍辛结构的存在产品的3-流形与圆，并限制哪些群体可以实现为基本组的代数曲线补。这个项目的目标是使用这些非交换的技术来找到与低维拓扑的三个特定领域相关的问题的答案：结Floer同调，代数曲线补的拓扑和结和谐群的结构。这个项目的目标是使用非交换代数来更好地理解打结曲线和曲面。纽结曲线曲面的研究在生物学和物理学中有着重要的应用。例如，DNA链是自然打结的，但必须解开才能复制。此外，了解打结的表面是了解宇宙形状的基础。许多数学家使用代数来更好地理解曲线和曲面如何打结，然而所考虑的代数类型通常是交换的。通过使用非交换代数，我们可以更精确地理解曲线和曲面如何打结。

A fundamental goal in low-dimensional topology is the classification of manifolds. The fundamental group of the manifold is an important algebraic invariant, but is rarely abelian. On the other hand, since the homology groups of the manifold are abelian, they can be classified easily.Unfortunately, a lot of the rich structure of the fundamental group is lost when considering the homology groups. Homology with local coefficients associates modules to the manifold. Since modules are abelian groups, distinguishing them is still tenable. Furthermore, since these are often modules over noncommutative rings, some of the rich structure of the fundamental group is retained. These homology modules, as well as linking forms defined on them, have been used to define algebraic invariants of knot complements, 3-manifolds, and algebraic curve complements. These have led to new results about knot concordance, obstructions to the existence of symplectic structures on the product of a 3-manifold with the circle, and restrictions on which groups can be realized as the fundamental group of an algebraic curve complement. The goal of this project is to use these noncommutative techniques to find answers to questions related to three particular areas of low-dimensional topology: knot Floer homology, the topology of algebraic curve complements, and the structure of the knot concordance group.The goal of this project is to use non-commutative algebra to better understand knotted curves and surfaces. The study of knotted curves and surfaces has important applications in biology and physics. For example, DNA strands are naturally knotted, but must unknot in order to replicate. Also understanding knotted surfaces is fundamental to understanding the shape of the universe. Many mathematicians have used algebra to better understand how curves and surfaces can knot, however the type of algebra considered is usually commutative. By employing non-commutative algebras, we can get a more refined understanding for how curves and surfaces can knot.