Contact structures and Floer homology on 3-manifolds with boundary

主要研究者旨在了解三维接触几何与三维流形不变量之间的相互作用。接触结构是起源于19世纪光学研究的几何对象；近年来，它们已经成为低维拓扑学的重要工具，提供了关于3维和4维空间（如我们的宇宙）形状的信息，以及在结理论中，它已经看到了与蛋白质折叠等主题的新兴联系。PI用来研究这些空间的许多工具，包括弗洛尔同调理论，都大量借鉴了数学物理，它们包含接触几何数据，这些数据提供了大量关于接触结构本身和它们所在空间的信息。拟议的研究将调查这些数据，并揭示它在数学的不同领域之间提供的联系，包括代数、拓扑和动力学。研究了规范理论和辛几何中出现的接触结构和3-流形的不变量，特别是带边界的3-流形的同调不变量。本项目的第一个目标是开发在缝合3流形的几个flower同调理论中接触不变量的拓扑学和辛几何的应用。这些潜在的应用包括拉格朗日调和的可计算障碍；在这种情况下推广了Weinstein猜想的证明，得到了接触3流形中Reeb轨道数的界；以及一个关于3流形的斯坦填充与其基本群表示的有趣猜想。第二个目标是建立不同的缝合线Floer同构理论之间的关系，这些理论对应的闭3流形不变量（Heegaard Floer同构、单极Floer同构和嵌入接触同构）现在都是同构的，并确定它们各自的接触不变量。第三个目标是研究Legendrian接触同调（LCH）和新的束理论Legendrian结不变量之间的新联系，并在此过程中将代数几何技术应用于接触几何问题，并有望理解LCH与经典结不变量的关系。

The principal investigator aims to understand the interplay between contact geometry in 3 dimensions and invariants of 3-manifolds. Contact structures are geometric objects which originated in the study of optics in the 19th century; in recent years they have become important tools in low-dimensional topology, providing information about the shapes of 3- and 4-dimensional spaces such as our universe, and in knot theory, which has seen emerging links to such topics as protein folding. Many of the tools which the PI uses to study such spaces, including Floer homology theories, draw heavily from mathematical physics, and they contain contact-geometric data which has provided a wealth of information about both the contact structures themselves and about the spaces in which they reside. The proposed research would investigate such data and uncover connections which it provides between disparate areas of mathematics, including algebra, topology, and dynamics.The PI intends to study invariants of contact structures and 3-manifolds, especially homological invariants of 3-manifolds with boundary, which arise from gauge theory and symplectic geometry. The first goal of this project is to develop applications to topology and to symplectic geometry of contact invariants in several Floer homology theories for sutured 3-manifolds. These potential applications include computable obstructions to Lagrangian concordances between Legendrian knots; bounds on the number of Reeb orbits in a contact 3-manifold, generalizing the proof of the Weinstein conjecture in this setting; and an intriguing conjecture relating Stein fillings of a 3-manifold to representations of its fundamental group. The second goal is to establish a relationship between different sutured Floer homology theories, whose corresponding closed 3-manifold invariants (Heegaard Floer homology, monopole Floer homology, and embedded contact homology) are now all known to be isomorphic, and to identify their respective contact invariants as well. The third goal is to investigate an emerging connection between Legendrian contact homology (LCH) and new sheaf-theoretic Legendrian knot invariants, and in doing so to apply algebro-geometric techniques to problems in contact geometry and hopefully understand the relationship of LCH to classical knot invariants.