Contact and Symplectic Structures and Holomorphic Curves

摘要建议：DMS-9802154首席研究员：赫尔穆特·霍费尔在这个项目中，霍费尔研究里布矢量场的动力学。主要的工具是一种特别适用的全纯曲线理论，这是首席研究员和他的同事近年来发展起来的。事实上，在一些重要的动力系统和合适的全纯曲线理论之间存在着如此密切但完全意想不到的关系，这一事实具有深刻的含义。它立即使关于一类重要动力系统的问题在原则上与最新的数学理论联系起来，如量子上同调、Gromov-Witten不变量，甚至在低维甚至与Seiberg-Witten不变量联系起来。四维空间中两条相交的全纯曲线总是具有正交数的事实表明，这些新方法对于三维空间上的Reeb矢量场是最有效的。该项目的主要目标将是开发一种用于构建三维Reeb流的全球截面表面及其一般化的数学机械。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归结为二维。下面的例子说明了Reeb动力学的重要性和范围。卫星在太阳、行星和月球引力作用下的运动在数学上被描述为里布矢量场的动力学。带电粒子在电场和磁场中运动时，如果磁场不太大或粒子的动能足够大，则用Reeb矢量场来描述。理想不可压缩流体的颗粒在某些管系中的运动，如果运动处于平衡(稳定)状态，则由Reeb矢量场描述，如果压力离恒定不太远。另外，许多描述能量保持不变的系统的数学物理演化方程有时可以用有限维动力系统来近似，而有限维动力系统则用Reeb矢量场来描述。如果一个人试图用计算机来求解这类方程，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.