Contact and Symplectic Structures and Holomorphic Curves

DMS-0102298赫尔穆特·H·霍费尔大多数拟议项目的主要工具是一种特别适用的全纯曲线理论，该理论是霍费尔博士和他的同事近年来开发的。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。它立即使得在原则上将关于大类动力系统的问题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。该项目的主要目标之一将是开发一种用于构造三维Reeb流的全球截面表面及其推广的数学机制。在实践中，这意味着对某一三维动力系统的研究可以有效地归结为二维。另一个目标是发展辛场理论。这种理论的特例是Gromov-Witten不变量和Floer同调。许多物理系统，如不可压缩理想流体的流动，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，都是所谓的动力系统的例子。动力系统的数学理论为理解它们的复杂行为和做出预测提供了工具。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，从拉格朗日和哈密尔顿的工作开始，几何工具已经发展到现代接触理论和辛几何。这些几何在连接动力系统、代数几何和光滑拓扑等数学领域中起着核心作用。这使得人们可以利用不同数学领域的强大工具来研究重要的动力系统类，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.