CAREER: Interactions between Floer Theory, Khovanov Homology, and Low-Dimensional Topology

这个研究项目涉及研究称为流形的三维和四维空间，以及它们的几何结构。 理解这些空间和结构是理解我们宇宙的形状和性质的核心。 例如，爱因斯坦通过广义相对论对引力的描述依赖于理解流形的数学工具，这些工具与该项目正在开发的工具没有太大区别。 事实上，许多正在研究的理论都有物理学的应用和起源。 该项目的一个主要目标是开发可能用于统一研究3维和4维流形的几个现有和重要工具的方法。 这种统一可能会揭示不同数学领域之间有趣的联系。 理解高维空间的一个重要方法是研究嵌入其中的低维空间;这个研究项目的大部分内容将集中在理解嵌入三维流形中的称为结的一维空间。 纽结理论是由化学家开尔文勋爵在19世纪80年代作为拓扑学的一个子领域提出的，它在数学之外还有许多其他有趣的应用-例如，研究某些酶如何改变DNA的打结。 该项目的目标之一是统一两个非常重要的，但表面上不同的，研究结的工具;有诱人的证据表明这两个理论之间的具体关系。 研究三维和四维流形的另一种方法是理解它们所支持的几何结构。 在这个项目中相当大的努力将进入了解接触结构的三维流形。 除了在数学上的应用，接触结构在经典力学、热力学、动力学系统和液晶研究中也很重要。 该项目还包含了几个明确的方向，为学生的研究，在本科和研究生水平，并将支持PI的教学和指导活动。 此外，该项目将通过创建一个包含背景和上下文的开放问题的Wiki存储库，为更广泛的社区服务。 PI还将利用该奖项的资金启动波士顿地区的研究生研讨会，并参加剑桥科学节上面向公众的数学推广活动。Floer理论彻底改变了低维拓扑和几何的研究。 这些弗洛尔理论中的许多理论似乎编码了相同的信息，表明辛几何和规范理论等领域之间的深层联系。 这些以及弗洛尔理论和表示论中的链接不变量之间的联系导致了低维拓扑学中的壮观结果，例如温斯坦猜想的证明。 尽管在理解这些联系方面取得了一些进展，但对它们还没有一个简单，统一的解释。 事实上，一个基本的开放问题，也是研究项目的一部分遥远的北极星，是将弗洛尔理论公理化。 一种方法涉及开发有边三维流形的不变量。 在这个项目中开发的加边的Border-Floer理论是新颖的，因为它也可以提供计算4-流形的不变量的方法。 这个项目的一个补充目标是阐明Floer理论和Khovanov同调之间的联系，Khovanov同调是由表示论激发的联系的不变量。 例如，这种联系有助于确定Khovanov同源性检测到解结。 该项目研究了一种新的方法来证明一个长期存在的猜想，该猜想将Khovanov同源性与结Floer同源性联系起来，这意味着Khovanov同源性也可以检测到8字形，三叶形和Hopf链接。 该项目还研究了一种新的方法，利用量子上同调对Floer理论的作用来研究3-流形的Heegaard分裂的丰富理论。 尽管Heegaard分裂在一些Floer理论的构建中起了作用，但这在以前还没有做过。 最后，该研究项目将采用新开发的规范理论接触不变量，提供新的见解勒让德结和拉格朗日和谐，并建立迄今未探索的接触几何和基本组之间的联系。

This research project involves studying 3- and 4-dimensional spaces called manifolds, and geometric structures on them. Understanding these spaces and structures is central to understanding the shape and properties of our universe. For example, Einstein's description of gravity via general relativity relies on mathematical tools for understanding manifolds that are not too different from those under development in this project. Indeed, many of the theories under study have applications and origins in physics. A major goal of the project is to develop methods that might be used to unify several existing and important tools for studying 3- and 4-dimensional manifolds. Such unification would likely reveal interesting connections between disparate fields of mathematics. An important method for understanding higher dimensional spaces is to study lower dimensional spaces that embed into them; much of this research project will be focused on understanding 1-dimensional spaces called knots embedded in 3-dimensional manifolds. Knot theory was initiated as a subfield of topology in the 1880's by the chemist Lord Kelvin, and it has many other interesting applications outside of mathematics -- to the study of how certain enzymes alter the knotting of DNA, for example. One of the project goals is to unify two very important, but prima facie different, tools for studying knots; there is tantalizing conjectural evidence of a concrete relationship between these two theories. Another means of studying 3- and 4-dimensional manifolds is by understanding the sorts of geometric structures they support. Considerable effort in this project will go into understanding contact structures on 3-dimensional manifolds. In addition to their applications in mathematics, contact structures are important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. This project also contains several explicit directions for student research, at both undergraduate and graduate levels, and will support the PI's teaching and mentoring activities. In addition, the project will serve the broader community by creating a Wiki repository of open problems, with background and context. The PI will also use the funds from this award to start a Boston-area graduate seminar, and to participate in math outreach for the general public at the Cambridge Science Festival.Floer theory has revolutionized the study of the topology and geometry in low dimensions. Many of these Floer theories appear to encode the same information, indicating deep connections between fields like symplectic geometry and gauge theory. These and connections between Floer theory and link invariants from representation theory have led to spectacular results in low-dimensional topology, such as proof of the Weinstein conjecture. Despite some progress in understanding these connections, there is not yet a simple, unifying explanation for them. Indeed, a fundamental open problem, and a distant lodestar for part of the research project, is to axiomatize Floer theory. One approach involves developing invariants of bordered 3-manifolds. The bordered monopole Floer theory under development in this project is novel in that it may provide methods for computing invariants of 4-manifolds as well. A complementary goal of this project is to elucidate links between Floer theory and Khovanov homology, an invariant of links motivated by representation theory. Such connections were instrumental, for example, in establishing that Khovanov homology detects the unknot. This project investigates a new approach to proving a long-standing conjecture relating Khovanov homology with knot Floer homology, which would imply, among other things, that Khovanov homology detects the figure eight, trefoil, and Hopf link as well. The project also investigates a novel approach for using actions of quantum cohomology on Floer theory to study the rich theory of Heegaard splittings of 3-manifolds. This has not been done before, despite the role Heegaard splittings play in the constructions of some Floer theories. Finally, the research project will employ newly-developed gauge-theoretic contact invariants to provide fresh insights into Legendrian knots and Lagrangian concordance, and to establish hitherto unexplored connections between contact geometry and the fundamental group.