Floer homology and contact and symplectic geometry

物理系统的所有可能构型的集合通常具有辛流形的结构。当人们限制到具有固定能量的构型时，通常会得到一个接触流形。因此，了解辛流形和接触流形的几何对于理解物理系统的动力学很重要。下面关于这种流形的两个几何问题特别有趣。首先，人们想要了解接触流形上的Reeb轨道，它对应于随时间重复的物理行为。其次，为了更好地理解不同辛流形之间的关系，人们希望了解一个辛流形何时可以辛嵌入到另一个辛流形中。接触同调是一个强大的工具，目前正在开发中，它可以应用于这两个问题。该项目将发展各种接触同调的基础、计算和应用。具体地说，嵌入接触同调(ECH)将被推广到三维接触流形上的函子和四维强辛余边线上。ECH和其他类型的接触同调的基础将在可能的情况下直接用全纯曲线来构造，以便将它们与几何更紧密地联系在一起。ECH容量(阻碍四维辛嵌入的数量不变量)将在更多的例子中计算，并与测地线流和哈密顿动力学有关。对于其他类型的接触同调，例如柱面接触同调和有理辛场理论，ECH容量的模拟将被构造和研究。这些新工具将被用来探索温斯坦猜想是否可以通过增加Reeb轨道数的下限或证明短Reeb轨道的存在来扩展。

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.