Invariants of bordered 3-manifolds and contact structures in Floer homology, connections with Khovanov homology, and applications

拟议的项目涉及研究三维和四维流形的几何结构。理解这些空间和结构是理解我们宏观宇宙形状的核心，PI打算研究的许多理论都有物理学的应用和起源。PI提案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。这样的统一可能会揭示出完全不同的数学领域之间有趣的联系。PI还计划使用其中一些工具以新的方式研究接触结构和结。两者都是研究三维和四维流形的探针。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，而纽结理论最近在DNA打结的研究中得到了许多应用。该计划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。在过去的几十年里，Floer理论已经彻底改变了拓扑和几何的光滑流形的三维和四维的研究。有多个不同的弗洛尔理论的3流形定义使用技术从不同的数学分支。令人惊讶的是，这些理论中有几个似乎给出了相同的信息，表明辛几何和规范理论等领域之间存在着深刻的联系。这些联系导致了美丽的结果，如陶布斯对温斯坦猜想的证明。尽管在理解这些联系方面取得了显著进展，但人们想知道是否存在更简单，统一的解释。事实上，一个基本的开放性问题和一个遥远的北极星为PI的建议是公理化弗洛尔理论。一种方法，这一点和相关问题的计算弗洛尔理论是发展不变量的边界3流形和手段计算不变量的一个封闭的流形从它的作品。第一个这样的不变量是由Lipshitz，Ozsvath和Szabo在Heegaard Floer homology中定义的。PI提出的加边界的Floer理论是新颖的，因为它也可以提供计算四维流形不变量的方法（它在HM-from，“主”版本的Floer同调中定义），并且可以移植到其他Floer理论（公理化方法的重要组成部分）。这一建议将支持这一理论的进一步发展。该提议的一个密切相关的方面涉及充实弗洛尔理论和霍瓦诺夫同调（Khovanov homology）之间的联系，霍瓦诺夫同调是一种由表示论激发的联系不变量。这种联系是有用的，例如，在克朗海默和Mrowka的著名证明，Khovanov同源检测unknot。通常这些联系出现在一个频谱序列的形式从霍瓦诺夫同源性和弗洛尔理论。这个项目的一个目标是更好地理解这些光谱序列。PI提出了一种组合计算其条款，并证明该序列是函的链接配边。希望使用这些组合方法和函子性来更好地理解3维和4维的Floer不变量。这个建议的最后一个补充方面涉及定义接触结构的新不变量。PI在最近的工作中已经这样做了，使用了瞬子和Escheriche Floer同源性。一个令人兴奋的未来项目是使用瞬子Floer不变量来揭示接触3-流形的Stein和辛填充与其基本群之间迄今未探索的联系。

The proposed project involves studying geometric structures on 3- and 4-dimensional manifolds. Understanding these spaces and structures is central to understanding the shape of our macroscopic universe, and many of the theories the PI intends to study have applications and origins in physics. A major goal of the PI's proposal is to develop methods which might be used to unify several existing and important tools for studying 3- and 4-dimensional manifolds. Such a unification would likely reveal interesting connections between quite disparate fields of mathematics. The PI also plans to use some of these tools to study contact structures and knots in new ways. Both serve as probes by which to study 3- and 4-dimensional manifolds. Contact structures are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals, while knot theory has found many recent applications in the study of DNA knotting. This proposal also contains several explicit directions for student research, at both undergraduate and graduate levels, and will support the PI's teaching and mentoring activities.Over the last couple decades, Floer theory has revolutionized the study of the topology and geometry of smooth manifolds in dimensions 3 and 4. There are multiple distinct Floer theories for 3-manifolds defined using techniques from quite different branches of mathematics. Amazingly several of these theories appear to give the same information, indicating deep connections between fields like symplectic geometry and gauge theory. These connections have lead to beautiful results like Taubes' proof of the Weinstein conjecture. Despite remarkable progress in understanding these connections, one wonders whether there is a simpler, unifying explanation. Indeed, a fundamental open problem and a distant lodestar for the PI's proposal is to axiomatize Floer theory. One approach to this and to the related problem of computing Floer theories is to develop invariants of bordered 3-manifolds and a means of computing the invariant of a closed manifold from those of its pieces. The first such invariant was defined by Lipshitz, Ozsvath, and Szabo in Heegaard Floer homology. The PI's proposed bordered monopole Floer theory is novel in that it may provide methods for computing invariants of 4-manifolds as well (it is defined in the HM-from, "master" version, of monopole Floer homology), and can likely be ported to other Floer theories (an important ingredient for an approach to axiomatization). This proposal will support the further development of this theory. A closely related aspect of the proposal involves fleshing out connections between Floer theory and Khovanov homology, an invariant of links motivated by representation theory. Such connections were instrumental, for example, in Kronheimer and Mrowka's celebrated proof that Khovanov homology detects the unknot. Often these connections arise in the form of a spectral sequence from Khovanov homology and Floer thoery. A goal of this project is to better understand these spectral sequences. The PI proposes a means for combinatorially computing their terms and for proving that the sequences are functorial with respect to link cobordisms. The hope is to use these combinatorial methods and functoriality to better understand Floer invariants in dimensions 3 and 4. A final, complementary aspect of this proposal involves defining new invariants of contact structures. The PI has done so in recent work, using instanton and monopole Floer homology. An exciting future project is to use the instanton Floer invariant to uncover hitherto unexplored links between the Stein and symplectic fillings of a contact 3-manifold and its fundamental group.