Topology and Contact and Symplectic Manifolds

辛流形是由经典力学中附加的结构构成的空间。接触流形在某种意义上是一种维度化简，并且历史上与光学和动力学的微分方程有联系。近年来，数学家们发现接触流形和辛流形的研究在我们对三维和四维空间的理解方面有很强的应用。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，著名的庞加莱猜想问球体的结构是否由一个相关的代数实体决定，即它的基本群；这个问题的四维版本仍然没有答案。有些令人惊讶的是，辛流形在回答这类问题方面发挥了重要作用。反过来，随着这一领域的发展，数学家们利用拓扑学、微分几何和物理学的工具来更好地理解接触流形和辛流形。本项目旨在利用接触流形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，同时开发新的工具来增加我们对接触流形和辛流形本身的理解。从某种意义上说,接触和辛拓扑桥梁刚度的黎曼几何和拓扑结构的灵活性,表现的特征:当地的灵活性和全球刚度。现代接触拓扑学始于20世纪80年代Bennequin的工作，并由Gromov和Eliashberg与辛拓扑学联系起来。吉鲁将拓扑学和三维接触几何紧密地结合在一起，通过将拓扑对象（称为打开的书分解的单一纤维束）与接触结构联系起来，以及描述与该接触结构兼容的所有打开的书的方法。该工具在与低维拓扑形成连接方面非常有效，允许构建新的接触不变量，对某些结进行手术表征，以及辛填充的分类等。打开的书还提供了两个新的接触结构的内在不变量：页面，打开的书中的纤维，和单一性，捆绑的粘合图。我们把一本兼容打开的书的最小格称为接触结构的页面格，这是一个非常有趣的不变量。如果页属为零，那么我们就可以得到关于接触结构的大量信息。如果页属不为零，则存在无限多的兼容的打开书籍，现有的描述它们的方法都远远不够有效，这使得确定页属是不可能的。事实上，是否存在最小页属大于1的接触结构是未知的。这个项目旨在简化这幅图，首先通过产生新的接触流形不变量，可以使用给定的打开的书有效地计算；其次，通过生成列出所有打开的书以及确定两个打开的书是否产生相同的接触结构的有效机制。

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.