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

DMS-0072202Viktor L.目前的建议侧重于几个长期项目，并继续主要研究者以前的工作由NSF赠款资助。第一个问题解决的proposalis的哈密尔顿塞弗特猜想，或者更具体地说，存在问题的哈密尔顿动力系统没有周期轨道上的一系列定期的能量水平。汉密尔顿塞弗特猜想是密切相关的下一组问题考虑的建议。这些问题属于辛拓扑学领域，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。 第二部分的建议包括一系列的问题，泊松几何。在这些问题中，例如，存在问题的等变泊松矩映射和建设泊松迹对应的叶子的辛叶理。一个通用的程序依赖于应用等变配边的研究哈密顿作用的紧凑groups. Hamilton动力系统的建议的结论部分的主题描述了许多类的物理过程中，可以忽略能量耗散。例如，天体力学中的行星运动和一些电动或磁动力学过程可以，而且通常是，被视为哈密顿动力系统。动力系统理论中的经典课题之一是周期轨道（即循环运动）的研究。周期运动是平衡后最简单也是最常见的运动类型，绝大多数哈密顿系统都具有周期轨道。第一个问题是构造不含周期轨道的Hamilton系统。这是一个相当重要的问题的理论哈密顿动力系统，因为这样的系统的例子将进一步推进我们的理解哈密顿动力学。下一个问题是关于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。这类哈密顿系统在物理学和力学的应用中自然出现。然而，最近在辛几何中开发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局限性，并导致新方法的发展。在提案中考虑的其他问题涉及古典力学系统和某些量子力学现象的几何性质之间的联系的研究。

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.