Floer homology and contact and symplectic geometry

一个物理系统的所有可能构型的集合通常具有辛流形的结构。当一个人被限制到具有固定能量的构型时，他通常会得到一个接触歧管。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是非常重要的。下面两个关于这种流形的几何问题特别有趣。首先，人们想要理解接触流形上的里布轨道，它对应于随时间重复的物理行为。其次，人们想要了解一个辛流形何时可以辛嵌入到另一个辛流形中，以便更好地理解不同辛流形之间的关系。接触同源性是一种强大的工具，目前正在开发中，可以应用于这两个问题。本项目将发展各种接触同源性的基础、计算和应用。特别是，嵌入式接触同调（ECH）将扩展到三维接触流形和四维强辛协矩阵上的函子。在可能的情况下，ECH和其他类型的接触同调的基础将直接用全纯曲线来构造，以便更紧密地将它们与几何联系起来。ECH能力（阻碍四维辛嵌入的数量不变量）将在更多的例子中计算，并与测地线流动和哈密顿动力学有关。本文将构建和研究其他类型的接触同调，如圆柱接触同调和有理辛场论的ECH容量的类似物。这些新工具将用于探索是否可以通过增加Reeb轨道数量的下界或证明短Reeb轨道的存在来扩展Weinstein猜想。

The set of all possible configurations of a physical system generally has the structure of a symplectic manifold. When one restricts to configurations with a fixed energy, one typically obtains a contact manifold. Understanding the geometry of symplectic manifolds and contact manifolds is thus important to understanding the dynamics of physical systems. The following two geometric questions about such manifolds are of particular interest. First, one would like to understand Reeb orbits on contact manifolds, which correspond to physical behavior which repeats over time. Second, one would like to understand when one symplectic manifold can be symplectically embedded into another, in order to better understand the relations between different symplectic manifolds. Contact homology is a powerful tool, currently under development, which can be applied to both of these questions.The project will develop the foundations, computation, and applications of various kinds of contact homology. In particular, embedded contact homology (ECH) will be extended to a functor on three-dimensional contact manifolds and four-dimensional strong symplectic cobordisms. The foundations of ECH and other kinds of contact homology will be constructed directly in terms of holomorphic curves when possible, in order to more closely relate them to geometry. ECH capacities (quantitative invariants which obstruct symplectic embeddings in four dimensions) will be computed in more examples and related to geodesic flows and Hamiltonian dynamics. Analogues of ECH capacities for other kinds of contact homology, such as cylindrical contact homology and rational symplectic field theory, will be constructed and studied. These new tools will be used to explore whether the Weinstein conjecture can be extended by increasing the lower bound on the number of Reeb orbits or proving the existence of short Reeb orbits.