Geometry and Dynamics of the group of Hamiltonian diffeomorphisms of a surface

表面哈密顿微分同胚群的几何与动力学

• 批准号: 0905911
• 负责人: Benson Farb
• 金额: $ 12.78万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905911&HistoricalAwards=false
• 关键词: Geometry; Dynamics; group; Hamiltonian; diffeomorphisms

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In this proposal, the PI plans to study various problems at the intersection of low dimensional topology and low dimensional dynamics on the one hand, and symplectic geometry and topology on the other hand. The purpose is to improve our understanding of certain objects, which appeared recently in the symplectic world (such as Hofer's metric) and to apply them to attack some old problems of geometrical and dynamical nature. Specific directions include the study of actions of higher rank lattices on surfaces, following Zimmer's program, of the simplicity of certain groups of transformations, and the existence of quasi-morphisms.The last twenty-five years have witnessed the birth of a remarkable mathematical field, namely symplectic topology. This field is connected to many areas of pure mathematics and modern physics. Some historical developments are associated to the name of Gromov, who introduced the concept of pseudo-holomorphic curve, as well as to the name of Hofer who introduced a remarkable new metric invariant in the classical field of Hamiltonian dynamics. In this project the PI plans to use these new objects and techniques, which are now already unavoidable, to establish new connections with other areas of pure mathematics such as geometric group theory.

该奖项是根据2009年《美国复苏与再投资法》（公法111-5）资助的。在该提案中，PI计划在低维拓扑和低维动态的交集中研究各种问题，另一方面是符号的几何学和拓扑。目的是提高我们对某些物体的理解，这些物体最近出现在符合性世界（例如Hofer的指标）中，并应用它们来攻击一些几何和动态性质的旧问题。具体的方向包括研究Zimmer的计划，某些转型群体的简单性以及准长态的存在的较高等级晶格在表面上的作用。过去的二十五年见证了一个了不起的数学领域的诞生，即符号拓扑。该领域与纯数学和现代物理学的许多领域有关。一些历史发展与格罗莫夫（Gromov）的名字有关，格罗莫夫（Gromov）介绍了伪塑形曲线的概念，以及霍弗（Hofer）的名字，霍弗（Hofer）在汉密尔顿动力学的古典领域引入了一个显着的新公制不变性。在该项目中，PI计划使用这些新对象和技术，这些新对象和技术现在已经不可避免地与其他纯数学领域（例如几何组理论）建立新的联系。