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尔顿公式中，而接触流形可以被看作是这样一种设置的扩展，以包括它如何随着时间的推移而发展，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、热力学和流体力学。然而，在现代意义上，研究辛流形的主要动机来自于它们在光滑4-流形研究中的地位，光滑4-流形是我们所生活的时空概念的推广。考虑这样一个空间沿着与辛结构已被证明允许一个定义非常强大的不变量的空间本身。由于它们可能非常复杂，这些空间通常通过切割成较小的，不太复杂的碎片，然后将它们粘在一起来研究。在辛设置中，包含在该构造中的信息主要由接触结构编码，沿着进行剪切和粘贴。这里的相关框架是一个“拓扑量子场论”，一个在数学的许多领域被证明是一个非常强大的想法的设置。在80年代和90年代，虽然很明显，某些接触流形表现出一种“灵活性”，使他们完全不适合他们在这个研究中的位置。这一发现，沿着这种结构在三维空间中的分类，是导致接触领域和辛拓扑学发展的主要重要发展之一。柔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比，它本身就适用于低维拓扑学的标准工具。由此产生的“灵活”与“僵化”现象的概念已成为该领域的核心。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，使每一方的机制和技术能够对另一方的问题产生影响。第一步涉及灵活性的“代数化”。建立在以前的工作的主要调查员，我们的目标是充分的概念，在语言的弗洛尔同源性，一个工具密切相关的上述几何研究的4-流形，并已被证明是特别有用的接触3-流形的研究。这是一个多方面的项目，其中涉及到一个完善的所谓的“接触类”的各种口味的弗洛尔同源性，并汇集了各种各样的工具，从一方面的拓扑和组合世界的分解3流形和其他更严格和代数世界的伪全纯曲线。这个项目将扩展到提供工具的算法检测的灵活性，以及定义一种测量刚性的方式。沿着，我们将探讨这个框架的关系，现有的概念“扭转”，和推广的框架到更高的维度，以及不同的几何形状（复杂与辛）。我们还将寻求应用的结果，以及以前的工作，应用程序有关的分类接触3-流形，特别是寻找例子的接触3-流形与任意的“支持genus”。

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''.