Contact and Symplectic Structures and Holomorphic Curves

摘要 提案：DMS-9802154主要研究者：Helmut霍费尔 在这个项目中，霍费尔研究了Reeb向量场的动力学。 的 主要工具是一个特别适用的全纯曲线理论， 由首席研究员和他的 近年来的同事。 事实上，有这样一个接近， 但完全出乎意料的是，一些重要的阶级之间的关系， 动力系统和一个合适的理论的全纯曲线， 深刻的影响。 原则上，它立即使我们能够 将关于一类重要的动力系统的问题与 最近的数学理论，如量子上同调，Gromov-Witten 不变量，在低维甚至到Seiberg-Witten不变量。 事实上，两个相交的全纯曲线在一个 四维空间总是有一个正的交集 这意味着这些新方法对于Reeb向量来说是最强大的 三维空间中的场。 这个项目的主要目标 将是发展一种数学机制， 三维Reeb流的整体断面， 其中的概括。 在实践中，这意味着研究 某些三维动力学系统可以有效地减少 变成二维的 Reeb动力学的重要性和范围由 下面的例子。 卫星的运动是在 描述了太阳、行星和月球的引力 在数学上可以看作是Reeb向量场的动力学。 带电 在电场和磁场中运动的粒子被描述为： 如果磁场不是太大， 粒子的动能足够大。 的运动 某些管道系统中理想不可压缩流体的微粒 如果运动处于平衡（稳定）状态，则由 如果压强离常数不远的话。 还有许多数学物理的演化方程，它们描述了 能量守恒的系统有时可以近似为 有限维动力系统，这是由一个Reeb 向量场 这种近似当然是必要的，如果 他试图用计算机来求解这些方程。

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.