Contact and Symplectic Structures and Holomorphic Curves

摘要提案：DMS-9802154项目负责人：Helmut Hofer在这个项目中，Hofer研究Reeb矢量场的动力学。主要的工具是一个特别适合的全纯曲线理论，这是由首席研究员和他的同事近年来发展起来的。在一类重要的动力系统和一种合适的全纯曲线理论之间存在着如此密切而又完全出乎意料的关系，这一事实具有深刻的意义。它立即使得，在原则上，将一类重要的动力系统的问题与最近的数学理论联系起来是可行的，比如量子上同调，Gromov-Witten不变量，在低维甚至与Seiberg-Witten不变量联系起来。四维空间中两条相交的全纯曲线的交点数总是正的这一事实表明，这些新方法对于三维空间中的Reeb向量场是最有效的。该项目的主要目标是开发一种数学机制，用于构建三维Reeb流的整体截面表面及其推广。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简化为二维。下面的例子说明了Reeb动力学的重要性和范围。在太阳、行星和月球的引力作用下，卫星的运动在数学上被描述为里布矢量场的动力学。在电场和磁场中运动的带电粒子，如果磁场不太大或粒子的动能足够大，则用里布矢量场来描述。理想不可压缩流体的粒子在某些管道系统中的运动，如果运动处于平衡（稳定）状态，则用里布矢量场描述，如果压力离恒定不远。同样，许多描述系统的数学物理演化方程，其中能量被保存，有时可以用有限维动力系统来近似，用Reeb向量场来描述。如果想用计算机求解这样的方程，这样的近似当然是必要的。

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.