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-0102298 Helmut H. HoferThe的主要工具，大多数拟议的项目是一个特别适应理论的全纯曲线，这已开发的博士霍费尔和他的同事在最近几年。在某些重要的动力系统类和一个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，这一事实具有深刻的意义。原则上，它立即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、Gromov-Witten不变量）联系起来成为可能，甚至在低维情况下也可以与Seiberg-Witten不变量联系起来。这个项目的主要目标之一将是开发一种数学机器，用于构造三维Reeb流及其推广的全球截面表面。 在实践中，这意味着某些三维动力系统的研究可以有效地减少到二维。 另一个目标是辛场论的发展。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，仅举几例，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。上面提到的具体例子是所谓的哈密顿性质。对于这样的系统，开始与工作的拉格朗日和汉密尔顿，几何工具已开发导致现代理论的接触和辛几何。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发挥着核心作用。这使得有可能采用强大的工具，从不同的数学领域的研究中的重要类别的动力系统，也使用的想法，从动力系统研究重要的内在问题，在其他领域的新方法。