Contact and Symplectic Structures and Holomorphic Curves

接触和辛结构以及全纯曲线

• 批准号: 0102298
• 负责人: Helmut Hofer
• 金额: $ 81.99万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-15 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102298&HistoricalAwards=false
• 关键词: Contact; Symplectic; Structures; Holomorphic; Curves

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.

DMS-0102298Helmut H. Hofer 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，该理论是由 Hofer 博士和他的同事近年来开发的。事实上，某些重要的动力系统与合适的全纯曲线理论之间存在着如此密切但完全出乎意料的关系，这一事实具有深远的意义。原则上，它使得将大类动力系统的问题与最近的数学理论（如量子上同调、格罗莫夫-维滕不变量，甚至低维中的塞伯格-维滕不变量）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 流的全局截面表面及其概括。 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。这种理论的特例是 Gromov-Witten 不变量和 Floer 同调。 许多物理系统，例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并允许做出预测的工具。上面提到的具体例子具有所谓的哈密顿性质。对于此类系统，从拉格朗日和汉密尔顿的工作开始，几何工具已经开发出来，导致了现代接触和辛几何理论。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发挥着核心作用。这使得在研究重要的动力系统类别时使用不同数学领域的强大工具成为可能，也可以利用动力系统的思想通过新方法研究其他领域的重要内在问题。