Mathematical Sciences: Symplectic and Poisson Geometry

数学科学：辛几何和泊松几何

• 批准号: 8702502
• 负责人: Jerrold Marsden
• 金额: $ 16.92万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702502&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symplectic; Poisson; Geometry

Marsden is one of the most active and productive workers in the field of symplectic geometry and dynamical systems. In joint work with many collaborators, he has developed very general differential geometric procedures for analyzing Hamiltonian systems with symmetry groups, and has applied these general ideas to an impressive range of physically interesting systems. Some outstanding characteristics of this work are the unified geometric and Lie theoretic framework it provides, the detailed working out of examples, and its applicability to field theories where one has infinite dimensional symmetry groups. Marsden and his collaborators at Berkeley and Arizona will carry out research into the following topics and their interconnections: Poisson geometry and the bundle picture; transverse structures, Lie algebras and singularities; differential equations in the complex domain; algebraic Poisson structures; geometry of bifurcations and normal forms of Hamiltonian systems; Hamiltonian chaos; relativistic field theories; modulation theory; Kac-Moody Lie algebras and integrable systems.

马斯登是辛几何和动力系统领域最活跃和最有成果的工作者之一。在与许多合作者的共同工作中，他开发了非常通用的微分几何程序，用于分析具有对称群的哈密顿系统，并将这些一般思想应用于一系列令人印象深刻的物理有趣系统。本工作的一些突出特点是它提供了统一的几何和李论框架，详细的例子，以及它在具有无限维对称群的场论中的适用性。马斯登和他在伯克利和亚利桑那的合作者将对以下主题及其相互联系进行研究：泊松几何和束图；横向结构，李代数和奇点；复域微分方程；代数泊松结构；哈密顿系统的分岔几何与范式哈密顿混乱;相对论场论；调制理论;Kac-Moody李代数与可积系统。