Rigidity and flexibility in contact topology

辛流形作为相空间出现在经典动力学的哈密顿公式中，接触流形可以被看作是这样一种设置的延伸，包括它如何随时间发展，或者是对固定能量系统的限制。因此，理解辛流形和接触流形的几何结构对于理解物理系统的动力学非常重要，并且在许多不同的领域都有应用，例如光学、热力学和流体动力学。然而，在现代意义上，研究辛流形的主要动机源于它们在光滑4流形研究中的地位，光滑4流形概括了我们所处的时空概念。考虑这样一个空间和一个辛结构已经被证明允许人们定义空间本身非常强大的不变量。由于这些空间可能非常复杂，因此研究这些空间的方法通常是将它们切割成更小、更不复杂的部分，然后将它们粘合在一起。在辛设置中，该构造中包含的信息主要由接触结构编码，并沿着接触结构执行剪切和粘贴。这里的相关框架是“拓扑量子场论”，这个理论在数学的许多领域被证明是一个非常强大的思想。在80年代和90年代，虽然很明显，某些接触歧管表现出一种“灵活性”，使它们完全不适合他们在这项研究中的位置。这一发现，以及这种结构在三维中的分类，是导致接触和辛拓扑学领域发展的重要发展之一。灵活性的研究在很大程度上是一个拓扑问题，与上面描述的基于代数-几何的设置相反，它可以使用低维拓扑的标准工具。由此产生的“柔性”与“刚性”现象的概念已经成为该领域的核心。本提案中概述的研究将侧重于这两个概念之间的“差距”，建立联系，通过这种联系，双方的机器和技术可以对另一方的问题产生影响。第一步涉及灵活性的“代数化”。在先前主要研究者工作的基础上，我们的目标是用Floer同调的语言充分描述这个概念，Floer同调是一种与上述4流形的几何研究密切相关的工具，并且在接触3流形的研究中被证明特别有用。这是一个多方面的项目，涉及到对各种类型的Floer同调的所谓“接触类”的改进，并汇集了各种各样的工具，一方面是3流形分解的拓扑和组合世界，另一方面是伪全纯曲线的刚性和代数世界。该项目将扩展到为灵活性的算法检测提供工具，以及定义测量刚性的方法。与此同时，我们将探索这个框架与现有“扭转”概念的关系，以及将框架推广到更高维度，以及不同的几何形状（复与辛）。我们还将寻求将结果以及先前的工作应用于有关接触3流形分类的应用，特别是寻找具有任意“支持属”的接触3流形的例子。

Symplectic manifolds arise as phase spaces in the Hamiltonian formulation of classical dynamics, and contact manifolds can be viewed as either an extension of such a setting to include how it progresses over time, or a restriction to a system of a fixed energy. Understanding the geometry of symplectic and contact manifolds is thus important to understanding the dynamics of physical systems, and has applications in many diverse fields, such as optics, thermodynamics, and hydrodynamics.In the modern sense, though, the main motivation for the study of symplectic manifolds derives from their place in the study of smooth 4-manifolds, spaces which generalize the notion of space-time in which we live. Considering such a space along with a symplectic structure has been shown to allow one to define very powerful invariants of the spaces themselves. As they can be extremely complicated, these spaces are often studied by cutting into smaller, less complicated pieces, then gluing these back together. In the symplectic setting, the information contained in this construction is primarily encoded by contact structures, along which the cutting and pasting are performed. The relevant framework here is that of a "topological quantum field theory", a set-up which has proved a hugely powerful idea in many areas of mathematics.In the 80's and 90's though it became clear that certain contact manifolds exhibit a kind of `flexibility', making them entirely unsuited to their place in this study. This discovery, along with the classification of such structures in 3-dimensions, was one of the major important developments leading to the growth of the field of contact and symplectic topology.The study of flexibility is very much a topological issue, lending itself to the standard tools of low-dimensional topology, in contrast to the algebraic-geometry-based setup described above. The resulting notions of `flexible' vs `rigid' phenomena have become central to the field. The research outlined in this proposal will focus on the ``gap'' between these two notions, building connections via which the machinery and techniques of each side can be brought to bear on the problems of the other. A first step concerns an ``algebraisation'' of flexibility. Building on previous work of the principle investigator, we aim to fully characterise the notion in the language of Floer homology, a tool closely related to the above-mentioned geometric study of 4-manifolds, and one which has proven particularly useful in the study of contact 3-manifolds. This is a multi-faceted project which involves a refinement of the so-called ``contact class'' of various flavors of Floer homology, and brings together a wide variety of tools from on the one hand the topological and combinatoric world of decompositions of 3-manifolds and on the other the more rigid and algebraic world of pseudo-holomorphic curves. This project will extend to provide tools for the algorithmic detection of flexibility, as well as defining a way of measuring rigidity.Along with this, we will explore the relation of this framework to existing notions of `torsion', and generalizations of the framework into higher dimensions, as well as differing geometries (complex vs symplectic). We will also seek to apply the results, as well as previous work, to applications concerning classification of contact 3-manifolds, in particular finding examples of contact 3-manifolds with arbitrary ``support genus''.