Contact and Symplectic Structures and Holomorphic Curves

接触和辛结构以及全纯曲线

• 批准号: 0603957
• 负责人: Helmut Hofer
• 金额: $ 112.3万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603957&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.

AbstractAward：DMS-0603957原理研究者：Helmut Hoferth symbletictic几何领域具有与其他有关学科的较大界面，例如代数几何，差分学（尤其是在较小的维度）和动态系统的少数名称。 Hofer博士的项目涉及符号几何形状的研究，其应用程序系统以及数学技术的发展，以解决在菲尔德引起的分析问题。该项目的一部分致力于核定场理论（SFT）的研究，该理论目前是符号不变性的最一般和最全面的理论。 另一部分开发了一种一般方法，用于研究某些否则椭圆形偏微分方程的类别。这些方法可能还应在非线性分析中具有其他应用。第三部分致力于该理论的载体系统。 其目的是基于浮点理论的建立和由于Hofer博士及其合作者而导致的有限能量食品理论的数学基础设施。这项研究对了解迭代性磁盘图的长期行为具有众多应用。 动态系统的数学理论提供了理解其复杂行为并允许进行预测的工具。 上面提到的特定示例是指的汉密尔顿性质，具有复杂的结构，以极端复杂的动态行为。稳定颗粒的颗粒的空白，或在星际旅程中发送拼空，或者理解不可压缩理想的理想流体的固定流动的动力学是问题，其数学基础是该项目中提出的数学基础。开发的某些方法可以应用于物理学中相关性的较大类别的较大类别的应用。