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Contact homology, dynamics, and embeddings

接触同源性、动力学和嵌入

基本信息

批准号：
2005437
负责人：
Michael Hutchings
金额：
$ 23.97万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005437&HistoricalAwards=false
关键词：
Contact homology dynamics embeddings

项目摘要

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.

这个NSF奖提供资金的一个项目，以研究某些问题的动力系统在辛几何。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。一个基本的目标是了解周期性轨道;这些对应于重复的行为，如一个单一的行星围绕一个星星旋转。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，以及是否随机配置接近周期。主要研究者还将研究与辛嵌入存在性相关的问题;辛嵌入是可以用来将不同动力系统相互联系起来的数学变换。为了研究这类动力学问题，将在三维流形的嵌入接触同调（ECH）理论以及其他更高维度的接触同调理论中开发数学工具。ECH的基础将扩展到Morse-Bott接触形式的情况。单位余切丛的ECH将被研究并与弦拓扑联系起来。谱不变量周期Floer同源（一个变种的ECH）将被开发和用于研究是否通用面积保持映射的表面有密集的周期轨道。ECH上的协边映射将被用来研究四维流形中的拉格朗日嵌入，并研究Arnold弦猜想的推广，即在给定的Legendrian结上存在端点为Reeb轨线。非等变和等变接触同调将在三维和更高的维度中进行研究。将比较这些不同类型的接触同调产生的辛容量。多面体上的组合Reeb动力学将被开发并用于进行计算机实验，以测试维泰博的猜想和其他有关Reeb动力学的理论。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。