Contact and Symplectic Structures and Holomorphic Curves

接触和辛结构以及全纯曲线

• 批准号: 9802154
• 负责人: Helmut Hofer
• 金额: $ 30.56万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802154&HistoricalAwards=false
• 关键词: Contact; Symplectic; Structures; Holomorphic; Curves

Abstract Proposal: DMS-9802154 Principal Investigator: Helmut Hofer In this project Hofer studies the dynamics of Reeb vector fields. The main tool is a particularly adapted theory of holomorphic curves, which has been developed by the principal investigator and his co-workers in recent years. The fact, that there is a such a close, but completely unexpected, relationship between some important class of dynamical systems and a suitable theory of holomorphic curves has deep implications. Immediately it makes it, in principle, feasible to relate questions about an important class of dynamical systems to recent mathematical theories like quantum cohomology, Gromov-Witten invariants, and in low dimensions even to Seiberg-Witten invariants. The fact, that two intersecting holomorphic curves in a four-dimensional space have always a positive intersection number implies that these new methods are most powerful for Reeb vector fields on a three-dimensional space. The main goal of this project will be the development of a mathematical machinery for constructing global surfaces of section for three-dimensional Reeb flows and generalisations thereof. In practice this means that the study of certain three-dimensional dynamical systems can effectively be reduced to two dimensions. The importance and scope of Reeb dynamics is illustrated by the following examples. The motion of a satellite in the presence of the gravitational forces of the sun, the planets and the moon is described mathematically as the dynamics of a Reeb vector field. A charged particle moving in an electrical and a magnetic field is described by a Reeb vector field if the magnetic field is not too large or the kinetic energy of the particle is sufficiently large. The motion of the particles of an ideal incompressible fluid in some system of pipes is, if the motion is in an equilibrium (steady) state, described by a Reeb vector field if the pressure is not too far f rom being constant. Also many evolution equations of mathematical physics, which describe systems, in which energy is preserved, can sometimes be approximated by finite-dimensional dynamical systems, which are described by a Reeb vector field. Such approximations are of course necessary if one tries to find solutions for such equations by using a computer.

摘要建议：DMS-9802154主要研究人员：Helmut Hofer在该项目中Hofer研究Reeb Vector Fields的动力学。 主要工具是一种特别改编的尸体曲线理论，近年来，主要研究人员及其同事已经开发了这种理论。 事实是，某些重要类别的动态系统与合适的霍明态曲线理论之间存在如此亲密但完全出乎意料的关系具有深刻的影响。 原则上，它立即使其可行，将有关一类动力学系统的问题与最近的数学理论相关联，例如量子共同体，gromov-witten不变性，甚至在低维度上，甚至与seiberg-witten的不变式。 事实是，在四维空间中的两个相交的全态曲线始终具有正相交的数字，这意味着这些新方法对于三维空间上的ReEB矢量场最强大。 该项目的主要目标是开发用于构建三维REEB流及其概括的部分的全局表面的数学机械。 在实践中，这意味着对某些三维动力系统的研究可以有效地降低到两个维度。 以下示例说明了REEB动力学的重要性和范围。 在太阳的引力，行星和月球存在的情况下，卫星的运动在数学上被描述为Reeb矢量场的动力学。 如果磁场不太大，或者粒子的动能足够大，则在电气和磁场中移动的带电粒子会用Reeb矢量场来描述。 如果在某些管道系统中，理想不可压缩流体的颗粒的运动是，如果运动处于平衡状态（稳定）状态，则由ReeB矢量场描述，如果压力不太远，则由ReeB矢量场描述。 同样，数学物理学的许多进化方程都描述了保留能量的系统，有时可以通过有限维动力学系统来近似，这些动力学系统由Reeb vector Field描述。 如果有人试图通过使用计算机来找到此类方程的解决方案，那么这种近似值当然是必要的。