Hamiltonian Dynamics and Pseudoholomorphic Curves

哈密​​顿动力学和伪全纯曲线

• 批准号: 1610452
• 负责人: Joel Fish
• 金额: $ 11.72万
• 依托单位: University of Massachusetts Boston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1610452&HistoricalAwards=false
• 关键词: Hamiltonian; Dynamics; Pseudoholomorphic; Curves

An example of a system studied in classical mechanics is the motion of a small satellite under the gravitational influence of planets. Historically, symplectic geometry grew out of studying situations such as this, in which the collection of all possible positions and momenta of the system (phase space) was seen to have the structure of a special space known as a symplectic manifold. Physical laws, like conservation of energy, then gave rise to models of the systems as dynamical systems on submanifolds of phase space corresponding to different energy levels. Modern symplectic methods have established deep connections between global properties of symplectic manifolds and the associated dynamics on a special class of energy levels, namely contact-type hypersurfaces, but have thus far proved to be of rather limited use for a general energy level. This project aims to rectify this deficiency and use global symplectic techniques to study dynamics on arbitrary energy levels by analyzing a generalized class of curves, known as feral pseudoholomorphic curves. This project concerns research at the interface of geometric analysis, symplectic topology, and dynamical systems. The key idea is the use of a certain newly defined class of infinite Hofer-energy pseudoholomorphic curves, namely feral curves, to study Hamiltonian flows on prescribed energy surfaces in symplectic manifolds of arbitrary dimension and certain volume-preserving flows on three-manifolds in general. More specifically, the aim is to study these curves and their properties and to use them to establish non-minimality of a wide range of volume preserving flows in dimension three. A further aim is to study the generic asymptotic limit of such feral curves and to determine whether or not symplectic field theory admits an extension to symplectic manifolds with generic smooth boundary. Successful completion of this research has the potential to solve the volume-preserving Gottschalk conjecture, broadly extend symplectic field theory, and possibly yield key new insights on low dimensional differential topology.

经典力学中研究系统的一个例子是小卫星在行星引力影响下的运动。历史上，辛几何起源于对这样的情况的研究，在这种情况下，系统(相空间)的所有可能的位置和动量的集合被视为具有被称为辛流形的特殊空间的结构。物理定律，如能量守恒定律，随后产生了相空间子流形上对应于不同能级的动力学系统模型。现代辛方法在一类特殊的能级，即接触型超曲面上，建立了辛流形的整体性质与相关动力学之间的深层联系，但到目前为止，对一般能级的使用是相当有限的。这个项目旨在弥补这一不足，并使用全局辛技术通过分析一类广义的曲线来研究任意能级上的动力学，这些曲线被称为野生伪全纯曲线。本项目涉及几何分析、辛拓扑和动力系统的界面研究。其核心思想是利用一类新定义的无限霍费尔-能量伪全纯曲线，即野生曲线，来研究任意维辛流形中给定能量面上的哈密顿流和一般三维流形上的某些保体积流。更具体地说，目的是研究这些曲线及其性质，并利用它们来建立三维大范围保容流动的非极小性。进一步的目的是研究这种野性曲线的一般渐近极限，并确定辛场理论是否允许将其推广到具有一般光滑边界的辛流形。这项研究的成功完成有可能解决保体积Gottschalk猜想，广泛推广辛场理论，并可能产生关于低维微分拓扑的关键新见解。