Periodic Orbits of Hamiltonian Systems, Cobordisms and Geometric Quantization, and Poisson Geometry

目前的提案侧重于几个长期项目，并继续由NSF资助的首席研究员以前的工作。第一个问题是哈密顿塞弗特猜想，更具体地说，是在规则能级序列上无周期轨道的哈密顿动力系统的存在性问题。哈密顿塞弗特猜想与提案中考虑的下一组问题密切相关。这些问题属于辛拓扑领域，涉及描述电荷在磁场中运动的哈密顿系统的周期轨道的存在性。第二部分包括泊松几何中的一系列问题。在这些问题中，例如，等变泊松矩映射的存在性问题和对应于辛叶理的叶的泊松迹的构造问题。利用等变协的一般规划来研究紧群的哈密顿作用，是本文结语部分的主题。哈密顿动力系统描述了许多可以忽略能量耗散的物理过程。例如，天体力学中的行星运动和一些电动力学或磁动力学过程可以而且通常被视为哈密顿动力学系统。动力系统理论中的一个经典课题是对周期轨道（即循环运动）的研究。周期运动是平衡之后最简单和最常见的运动。据信，绝大多数的哈密顿系统都有周期轨道。第一个问题是在没有周期轨道的情况下构造哈密顿系统。这是一个对哈密顿动力系统理论相当重要的问题，因为这种系统的例子将进一步推进我们对哈密顿动力学的理解。下一个问题涉及描述电荷在磁场中的运动的哈密顿系统的周期轨道的存在性。这类哈密顿系统自然出现在物理学和力学的应用中。然而，最近在辛几何中发展的非常强大的一般方法很少适用于这类系统。对这些系统的研究应该扩展现有方法的限制，并导致新方法的发展。提案中考虑的其他问题涉及经典力学系统的几何性质与某些量子力学现象之间的联系的研究。

DMS-0072202Viktor L. GinzburgThe present proposal focuses on several long-term projectsand continues principal investigator's previous work funded by an NSF grant. The first question addressed in the proposalis the Hamiltonian Seifert conjecture or, more specifically, the existence problem for Hamiltonian dynamical systems without periodic orbits on a sequence of regular energy levels. The Hamiltonian Seifert conjecture is closely related to the next group of questions considered in the proposal. These questions lie in the area of symplectic topology and concern the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. The second part of the proposal includes a series of problems in Poisson geometry. Among these problems are, for example, the existence questions for equivariant Poisson moment maps and the construction of Poisson traces corresponding to the leaves of the symplectic foliation. A general program relying on applications of equivariant cobordisms to the study of Hamiltonian actions of compact groups is the subject of the concluding part of the proposal.Hamiltonian dynamical systems describe many classes of physical processes in which dissipation of energy can be neglected. For example, planetary motion in celestial mechanics and some electro- or magneto-dynamical processes can be, and usually are, treated as Hamiltonian dynamical systems. One of the classical subjects in the theory of dynamical systems is the study of periodic orbits (i.e. cyclic motions). Periodic motion is the simplest and most common type of motion after equilibrium.It is believed that a vast majority of Hamiltonian systems have periodic orbits. The first problem addressed in theproposal is the construction of Hamiltonian systems without periodic orbits. This is a question of considerable importance for the theory of Hamiltonian dynamical systems because examples of such systems would further advance our understanding of Hamiltonian dynamics. The next problem concerns the existence of periodic orbits for Hamiltonian systems describing the motion of a charge in a magnetic field. This class of Hamiltonian systems naturally arises in applications in physics and mechanics. However, few of the extremely powerful general methods that have been recently developed in symplectic geometry are applicable to this class of systems. The investigation of these systems should extend the limits of existing methods and result in the development of novel ones. Other problems considered in the proposal concern the study of connections betweengeometrical properties of classical-mechanical systems and certain quantum-mechanical phenomena.