Contact and Symplectic Structures and Holomorphic Curves

大多数被提议的项目的主要工具是一个特别适合的全纯曲线理论，这是由Hofer博士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，将大的动力系统类的问题与最近的数学理论，如量子上同调，Gromov-Witten不变量，甚至在低维中与Seiberg-Witten不变量联系起来是可行的。该项目的主要目标之一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。另一个目标是发展辛场论。这种理论的特殊情况是Gromov-Witten不变量和Floer同调。许多物理系统，如不可压缩的理想流体的流动，卫星在天体引力作用下的运动，或带电粒子在磁场中的运动，等等，都是所谓动力系统的例子。动力系统的数学理论提供了理解其复杂行为的工具，并允许进行预测。上面提到的特殊例子具有所谓的哈密顿性质。对于这样的系统，从拉格朗日和汉密尔顿的工作开始，几何工具被开发出来，导致了现代接触理论和辛几何。这些几何在连接数学领域，如动力系统、代数几何和光滑拓扑中起着核心作用。这使得我们可以利用来自不同数学领域的强大工具来研究重要的动力系统类别，也可以利用动力系统的思想以新的方法来研究其他领域的重要内在问题。

DMS-0102298Helmut H. HoferThe main tool for most of the proposed projects is a particularly adapted theory of holomorphic curves, which has been developed by Dr. Hofer and his co-workers in recent years. The fact, that there is 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 large classes of dynamical systems to recent mathematical theories like quantum cohomology, Gromov-Witten invariants, and in low dimensions even to Seiberg-Witten invariants. One of the main goals of this project will be the development of a mathematical machinery for constructing global surfaces of section for three-dimensional Reeb flows and generalizations thereof. In practice this means that the study of certain three-dimensional dynamical system can effectively be reduced to two dimensions. The other goal is the development of a symplectic field theory. Particular cases of such a theory are Gromov-Witten invariants and Floer Homology. Many physical systems like the flow of an incompressibleideal fluid, the movement of a satellite under the gravitational forces of celestial bodies, or the movement of charged particles in a magnetic field, to name a few, are examples of so called dynamical systems. The mathematical theory of dynamical systems provides tools to understand their complex behavior and allowsto make predictions. The particular examples mentioned above are of so-called Hamiltonian nature. For such systems, beginning with the work of Lagrange and Hamilton, geometric tools have been developed leading to the modern theory of contact and symplectic geometry. These geometries play a central role in connecting mathematical areas like dynamical systems, algebraic geometry and smooth topology. This makes it possible to employ powerful tools from different mathematical areas in the study of important classes of dynamical systems and also to use ideas from dynamical systems to study important intrinsic questions in other fields by new methods.