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

这个研究项目包括研究被称为流形的三维和四维空间，以及它们上面的几何结构。理解这些空间和结构对于理解宇宙的形状和性质至关重要。例如，爱因斯坦通过广义相对论对引力的描述依赖于数学工具来理解流形，这些工具与本项目中正在开发的工具并没有太大的不同。事实上，许多正在研究的理论在物理学中都有应用和起源。该项目的一个主要目标是开发可以用于统一研究三维和四维流形的几种现有和重要工具的方法。这样的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将集中在理解被称为嵌入在三维流形中的结的一维空间上。打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分支发起的，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。项目目标之一是统一两个非常重要但表面上不同的研究结的工具；有诱人的推测证据表明这两种理论之间存在着具体的关系。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，相当多的努力将用于理解三维流形上的接触结构。除了在数学上的应用外，接触结构在经典力学、热力学、动力系统和液晶研究中也很重要。该项目还包括本科生和研究生阶段的几个明确的学生研究方向，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存储库来服务于更广泛的社区。PI还将利用这笔奖金在波士顿地区举办研究生研讨会，并在剑桥科学节（Cambridge Science Festival）上参与面向公众的数学推广活动。花理论对低维拓扑和几何的研究产生了革命性的影响。许多Floer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存在着深刻的联系。Floer理论和表示理论中的链接不变量之间的这些联系在低维拓扑中导致了惊人的结果，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，但还没有一个简单、统一的解释。事实上，一个基本的开放问题，以及部分研究项目的一个遥远的方向，就是使弗洛尔理论公理化。一种方法涉及开发有边3流形的不变量。本课题所发展的有边单极子Floer理论的新颖之处在于它也为计算4流形的不变量提供了方法。这个项目的一个补充目标是阐明弗洛尔理论和霍瓦诺夫同调之间的联系，霍瓦诺夫同调是由表征理论激发的联系的不变量。例如，这种联系有助于确立霍瓦诺夫同源性检测解结。本项目研究了一种新的方法来证明一个长期存在的关于Khovanov同源性与结花同源性的猜想，这将意味着，除其他事项外，Khovanov同源性也能检测到数字8、三叶草和Hopf链接。研究了利用花理论上的量子上同调作用来研究3流形heegard分裂的富理论的新方法。尽管heegard分裂在一些弗洛尔理论的构建中发挥了作用，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，为Legendrian结和Lagrangian调和提供新的见解，并建立迄今为止尚未探索的接触几何与基本群之间的联系。

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.