Contact homology, dynamics, and embeddings

这个NSF奖为一个研究辛几何动力系统中某些问题的项目提供资金。这些动力系统是物理过程的数学模型，例如太阳系中行星的运动。一个基本目标是了解周期轨道；这与重复的行为相对应，比如一颗行星围绕一颗恒星旋转。特别是，对于一个给定的动力系统，重要的是要了解有多少个周期轨道存在，以及一个随机构型是否接近周期构型。首席研究员还将研究辛嵌入是否存在的相关问题；这些是数学转换，可以用来将不同的动力系统相互联系起来。为了研究这类动力学问题，将发展三维流形的内嵌接触同调理论和其他高维接触同调理论的数学工具。ECH的基础将扩展到莫尔斯-博特接触形式的情况。将研究单位共切束的ECH，并将其与弦的拓扑结构联系起来。周期Floer同调中的谱不变量（ECH的一种变体）将被开发并用于研究曲面的一般保面积映射是否具有密集的周期轨道。ECH上的协同映射将用于研究四流形中的拉格朗日嵌入，并研究关于端点在给定Legendrian结上的Reeb轨迹存在性的Arnold弦猜想的推广。非等变和等变接触同调将在三维及更高的维度上进行研究。我们将比较这些不同类型的接触同源体所产生的辛容量。将开发多面体上的组合Reeb动力学，并将其用于进行计算机实验，以测试Viterbo的猜想和其他有关Reeb动力学的猜想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This NSF award provides funding for a project to study certain questions in dynamical systems in symplectic geometry. These dynamical systems are mathematical models for physical processes such as the motion of the planets in the solar system. One basic goal is to understand periodic orbits; these correspond to repeating behavior such as a single planet revolving around a star. In particular, for a given dynamical system, it is important to understand how many periodic orbits exist, and whether or not a random configuration is close to a periodic one. The principal investigator will also study related questions about the existence of symplectic embeddings; these are mathematical transformations that can be used to relate different dynamical systems to each other.In order to study these kinds of dynamical questions, mathematical tools will be developed in the theory of embedded contact homology (ECH) of three-manifolds, and other kinds of contact homology in higher dimensions. The foundations of ECH will be extended to the case of Morse-Bott contact forms. ECH of unit cotangent bundles will be studied and related to string topology. Spectral invariants in periodic Floer homology (a variant of ECH) will be developed and used to study whether or not generic area-preserving maps of surfaces have dense periodic orbits. Cobordism maps on ECH will be used to study Lagrangian embeddings in four-manifolds, and to investigate generalizations of the Arnold chord conjecture on the existence of Reeb trajectories with ends on a given Legendrian knot. Nonequivariant and equivariant contact homology will be studied in three and higher dimensions. Symplectic capacities arising from these different kinds of contact homologies will be compared. Combinatorial Reeb dynamics on polytopes will be developed and used to perform computer experiments to test Viterbo's conjecture and other related conjectures about Reeb dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.