Symplectic topology of Hamiltonian systems with infinitely many periodic orbits

所提出的研究的主要目的是研究各种哈密顿动力系统存在无限多个周期轨道的现象，并理解允许有限多个周期轨道的系统的本质。该提案由几个相互关联的项目组成，这些项目针对某些哈密尔顿微分同胚类以及特定的哈密顿系统，例如磁流，解决这些问题。PI将通过使用辛拓扑中的方法，包括Floer和量子同调技术，全纯曲线，谱不变量，Ljusternik-Schnirelman理论，以及微分几何中的方法，如h-原理来解决这些问题。PI所使用的技术还具有超越无穷多个周期轨道存在问题的应用。特别是，关于Poincaré常返性、Reeb流和余迷向辛拓扑的项目在很大程度上借鉴了她最近关于周期轨道的工作。这些方案在保测动力系统、经典动力系统以及辛拓扑中的嵌入问题上都有应用。哈密顿系统构成了一大类可以忽略耗散力的物理系统。例如，天体力学中的行星运动、不可压缩理想流体的流动以及带电粒子在磁场中的运动通常被视为哈密顿系统。这类系统的一个普遍但不是普遍的特征是，它们往往有许多周期轨道。与周期运动相对应，这是平衡后最简单的动力学现象，研究系统的周期轨道对于理解其整体行为至关重要。仅举几个例子，周期轨道的知识在天文学、流体动力学(例如湍流的统计)中是至关重要的，或者可以用来理解解的大时间稳定性。除了最简单的情况外，在所有情况下，确定周期轨道的存在通常需要先进和强大的数学工具。对于一类广泛的哈密顿系统，周期轨道的数目是已知的无限的，这被认为是许多哈密顿系统的情况，但不是所有的情况。讨论了哈密顿动力系统在不同环境下无穷多个周期轨道的存在性问题，以及PI用来解决这一问题的技巧在其他相关问题上的应用。提案最后部分的项目涉及某一类空间，例如，在研究具有对称性的哈密顿系统时出现的空间。拟议的工作与数学物理、几何和量子力学有关，并具有潜在的应用价值。

The primary goal of the proposed research is to investigate the phenomenon of existence of infinitely many periodic orbits for a variety of Hamiltonian dynamical systems and to understand the nature of the systems admitting finitely many periodic orbits. The proposal comprises several interconnected projects addressing these questions for certain classes of Hamiltonian diffeomorphisms and also for specific Hamiltonian systems such as magnetic flows. The PI will tackle these problems by employing methods from symplectic topology including Floer and quantum homological techniques, holomorphic curves, spectral invariants, Ljusternik-Schnirelman theory, as well as methods from differential geometry such as h-principles. The techniques utilized by the PI also have applications beyond the question of existence of infinitely many periodic orbits. In particular, the projects concerning the Poincaré recurrence, the Reeb flows and the coisotropic symplectic topology draw heavily on her recent works concerning periodic orbits. These projects have applications to measure-preserving and classical dynamical systems, and to some embedding problems in symplectic topology.Hamiltonian systems constitute a broad class of physical systems where dissipative forces can be disregarded. For example, the planetary motion in celestial mechanics, the flow of an incompressible ideal fluid, and the motion of a charged particle in a magnetic field are usually treated as Hamiltonian systems. One general, but not universal, feature of such systems is that they tend to have numerous periodic orbits. Corresponding to the cyclic motion, this is the simplest dynamical phenomenon after equilibrium, and an investigation of periodic orbits of a system is crucial in understanding its global behavior. To give but a few applications, the knowledge of periodic orbits is crucial in astronomy, fluid dynamics (e.g., statistics of turbulent flow) or can be used to understand stability of solutions for large times. In all but the simplest cases, establishing existence of periodic orbits often requires advanced and powerful mathematical tools. For a broad class of Hamiltonian systems, the number of periodic orbits is known to be infinite and this is thought to be the case for many, but not all, Hamiltonian systems. The proposal focuses on the problem of existence of infinitely many periodic orbits for Hamiltonian dynamical systems in a variety of settings and on applications of the techniques used by the PI to attack this problem to some other related questions. The projects in the last part of the proposal concern a certain class of spaces which arise, for instance, in the study of Hamiltonian systems with symmetries. The proposed work is related to and has potential applications in mathematical physics, and geometric and quantum mechanics.