Topology and Contact and Symplectic Manifolds

辛流形是一种具有来自经典力学的额外结构的空间。接触流形在某种意义上是一种维度简化，并且在历史上与光学和动力学的微分方程有联系。近年来，数学家们发现接触和辛流形的研究对我们理解三维和四维空间有很强的应用。拓扑学的一个基本目标是理解流形的某些代数简化在多大程度上决定了流形本身。例如，著名的庞加莱猜想（Poincare Conjecture）询问球体的结构是否由称为其基本群的相关代数实体决定;这个问题的四维版本仍然没有答案。有些令人惊讶的是，辛流形在回答这些问题方面发挥了重要作用。反过来，随着该领域的发展，数学家们使用了拓扑学，微分几何和物理学的工具，目的是更好地理解接触和辛流形。该项目旨在使用接触和辛流形研究的工具来进一步理解三维和四维空间，并开发新的工具来增加我们对接触和辛流形本身的理解。从某种意义上说，切触拓扑和辛拓扑是黎曼几何的刚性和拓扑的柔性的桥梁，同时具有局部柔性和整体刚性的特点。现代接触拓扑开始于20世纪80年代与Bennequin的工作，并连接到辛拓扑的格罗莫夫和Eliashberg。吉鲁把拓扑学和接触几何学在三维空间中紧密地联系在一起，他把一个拓扑对象，一个被称为开卷分解的奇异纤维丛，与一个接触结构联系在一起，并提出了一种描述所有与该接触结构兼容的开卷的方法。这个工具在形成低维拓扑的连接方面非常有效，允许构建新的接触不变量，某些结的手术特征，以及辛填充的分类等。打开的书还提供了接触结构的两个新的内在不变量：页面，打开的书中的纤维，以及单值性，束的粘合映射。我们称相容开书的最小亏格为接触结构的页亏格，它是一个非常有趣的不变量。如果页面亏格为零，那么我们可以对接触结构进行大量的描述。如果页属不为零，则存在无限多个兼容的打开的书，并且用于描述它们的现有方法远远不是有效的，这使得确定页属是不可能的。实际上，不知道是否存在最小页属大于1的接触结构。这个项目的目的是简化这幅图，首先通过产生新的接触流形的不变量，可以有效地计算使用一个给定的打开的书;第二，通过产生有效的机制，列出所有打开的书，以及确定是否两个打开的书产生相同的接触结构。

Symplectic manifolds are spaces equipped with an additional structure coming from classical mechanics. Contact manifolds are in some sense a dimensional simplification, and have historically had connections to the differential equations of optics and dynamics. In recent years, mathematicians have found strong applications of the study of contact and symplectic manifolds to our understanding of three- and four-dimensional spaces. One foundational goal in topology is to understand the extent certain algebraic simplifications of a manifold determine the manifold itself. For example, the famous Poincare Conjecture asks whether the structure of a sphere is determined by a related algebraic entity known as its fundamental group; a four dimensional version of this question is still unanswered. Somewhat surprisingly, symplectic manifolds have played a strong role in answering such questions. In turn, as the field has progressed, mathematicians have used tools from topology, differential geometry and physics with the goal of better understanding contact and symplectic manifolds. This project aims to both use the tools from the study of contact and symplectic manifolds to further our understanding of three- and four-dimensional spaces, as well as to develop new tools to increase our understanding of contact and symplectic manifolds themselves. In one sense, contact and symplectic topology bridges the rigidity of Riemannian geometry and the flexibility of topology, showing traits of both: local flexibility and global rigidity. Modern contact topology began in the 1980s with Bennequin's work and was connected to symplectic topology by Gromov and Eliashberg. Giroux brought topology and contact geometry in 3-dimensions closely together by associating a topological object, a singular fiber bundle called an open book decomposition, to a contact structure, as well as a method for describing all open books compatible with that contact structure. This tool has been extremely effective at forming connections with low-dimensional topology, allowing for the construction of new contact invariants, surgery characterization of certain knots, and the classification of symplectic fillings, among much else. Open books additionally provide two new intrinsic invariants of the contact structure: the page, a fiber in the open book, and the monodromy, the gluing map of the bundle. We call the minimal genus of a compatible open book the page genus of the contact structure, and it is an extraordinarily interesting invariant. If the page genus is zero, then we can say a tremendous amount about the contact structure. If the page genus is not zero, then there are infinitely many compatible open books and the existing methods for describing them all are far from effective, which makes determining the page genus impossible. Indeed, it is unknown whether there are contact structures with minimal page genus greater than one. This project aims to simplify this picture, first by producing new invariants of contact manifolds that can be effectively calculated using a given open book; and second by producing effective mechanisms for listing all open books as well as determining whether two open books yield the same contact structure.