Topology and Contact and Symplectic Manifolds

辛流形是具有来自经典力学的附加结构的空间。接触流形在某种意义上是一种维度简化，历史上与光学和动力学的微分方程式有联系。近年来，数学家们发现接触流形和辛流形的研究对我们理解三维和四维空间具有很强的应用价值。拓扑学的一个基本目标是了解流形的某些代数简化决定流形本身的程度。例如，著名的庞加莱猜想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很大的作用。反过来，随着这一领域的发展，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。这个项目的目的既是利用接触和辛流形的研究工具来加深我们对三维和四维空间的理解，也是为了开发新的工具来增加我们对接触和辛流形本身的理解。在某种意义上，接触拓扑和辛拓扑连接了黎曼几何的刚性和拓扑的灵活性，表现出两者的特点：局部柔性和全局刚性。现代接触拓扑学始于20世纪80年代Bennequin的工作，Gromov和Eliashberg将其与辛拓扑联系起来。Giroux通过将拓扑对象、称为开卷分解的单一纤维丛与触点结构相关联，以及描述与该触点结构兼容的所有开卷的方法，将3维中的拓扑和触点几何紧密地结合在一起。这个工具在形成与低维拓扑的连接方面非常有效，允许构造新的接触不变量，某些结的外科特征，以及辛填充的分类等等。翻开的书还提供了接触结构的两个新的内在不变量：页面，打开书中的一根纤维，以及Mondromy，捆绑的粘合映射。我们称相容开卷的最小亏格为联系结构的页亏格，它是一个非常有趣的不变量。如果页面属数为零，那么我们可以说大量关于联系结构的信息。如果页面亏格不是零，那么有无限多的兼容打开的书籍，并且现有的描述它们的方法远远不能有效地描述它们，这使得确定页面亏格是不可能的。事实上，是否存在最小页面属大于1的联系结构是未知的。该项目旨在简化这一图景，首先通过产生可以使用给定的开卷有效计算的接触流形的新不变量；其次通过产生有效的机制来列出所有开卷以及确定两个开卷是否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.