Contact and Symplectic Structures and Holomorphic Curves

AbstractAward：DMS-0603957首席研究员：赫尔穆特·霍弗辛几何领域与其他数学学科有着很大的接口，如代数几何、微分拓扑（特别是在小维度上）和动力系统等。霍费尔博士的项目涉及辛几何的基础方面的研究，其在动力系统中的应用，以及解决该领域中出现的分析问题的几何技术的发展。本项目的一部分是致力于辛场论（SFT）的研究，这是目前最普遍和最全面的辛不变量理论。 另一部分发展了一个一般的方法来研究某些类的非线性椭圆型偏微分方程，因为他们发生在SFT。这些方法在非线性分析中也有潜在的应用。第三部分致力于理论的应用到动力系统。 其目的是发展数学基础设施，其基础是弗洛尔理论和霍费尔博士及其合作者的有限能量叶理论的结合。本研究旨在了解迭代面积保持圆盘映射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，带电粒子在磁场中的运动，等等，都是所谓的动力系统的例子。 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。 上面提到的特殊例子是所谓的哈密顿性质，具有复杂的结构，导致极端复杂的动力学行为。在粒子加速器中稳定粒子束，或者在星际旅行中发送探测器，或者理解不可压缩理想流体的静止流的动力学，这些问题的数学基础都被这个项目中提出的研究所触及。一些方法的开发潜在的应用到更大类的偏微分方程的相关物理。

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.