Contact and Symplectic Structures and Holomorphic Curves

接触和辛结构以及全纯曲线

• 批准号: 1047602
• 负责人: Helmut Hofer
• 金额: $ 33.05万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1047602&HistoricalAwards=false
• 关键词: Contact; Symplectic; Structures; Holomorphic; Curves

AbstractAward: DMS-0603957Principal Investigator: Helmut HoferThe field of symplectic geometry has a large interface to othermathematical disciplines, like algebraic geometry, differentialtopology (particularly in small dimensions) and dynamical systemsto name a few. Dr. Hofer's project is concerned with the study offundamental aspects of symplectic geometry, its applications todynamical systems, as well as the development of mathematicaltechnology to address analytical problems arising in thefield. One part of the project is devoted to the study ofSymplectic Field Theory (SFT) which currently is the most generaland most comprehensive theory of symplectic invariants. Anotherpart develops a general approach for studying certain classes ofnonlinear elliptic partial differential equations as they occurin SFT. These methods potentially should have other applicationsin nonlinear analysis as well. A third part is devoted to theapplications of the theory to dynamical systems. The aim is thedevelopment of mathematical infrastructure, based on acombination of Floer theory and the theory of finite energyfoliations due to Dr. Hofer and his collaborators. This researchaims at the understanding of long-term behavior of iteratedarea-preserving disk maps with its numerous applications.Many physical systems like the flow of an incompressible idealfluid, the movement of a satellite under the gravitational forcesof celestial bodies, or the movement of charged particles in amagnetic field, to name a few, are examples of so calleddynamical systems. The mathematical theory of dynamical systemsprovides tools to understand their complex behavior and allows tomake predictions. The particular examples mentioned above are ofso-called Hamiltonian nature and have an intricate structureleading to extreme complicated dynamical behavior. Stabilizing abeam of particles in a partic= le accelerators, or sending aprobe on an interstellar journey, or understanding the dynamicsof a stationary flow of an incompressible ideal fluid areproblems whose mathematical underpinnings are touched by theresearch proposed in this project. Some of the methods developedpotentially have application to larger classes of partialdifferential equations of relevance in physics.

摘要奖：DMS-0603957 首席研究员：Helmut Hofer 辛几何领域与其他数学学科有很大的接口，例如代数几何、微分拓扑（特别是在小维中）和动力系统等等。 Hofer 博士的项目涉及辛几何的基本方面、其在动力系统中的应用以及数学技术的发展以解决该领域中出现的分析问题的研究。该项目的一部分致力于研究辛场论（SFT），它是目前最普遍、最全面的辛不变量理论。 另一部分开发了一种通用方法来研究 SFT 中出现的某些类别的非线性椭圆偏微分方程。这些方法在非线性分析中也可能有其他应用。第三部分致力于该理论在动力系统中的应用。 其目标是基于 Floer 理论和 Hofer 博士及其合作者的有限能量叶理理论的数学基础设施的发展。这项研究旨在了解迭代保面积圆盘图的长期行为及其众多应用。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的运动仅举几例，磁场中就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进行预测的工具。 上面提到的具体例子具有所谓的哈密顿性质，并且具有导致极其复杂的动力学行为的复杂结构。稳定粒子加速器中的粒子束，或在星际旅行中发送探测器，或理解不可压缩理想流体的静态流动的动力学，都是本项目提出的研究触及其数学基础的问题。所开发的一些方法可能适用于物理学中相关的更大类偏微分方程。