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年美国复苏和再投资法案（公法111-5）资助的。在该提案中，PI计划一方面研究低维拓扑和低维动力学交叉点的各种问题，另一方面研究辛几何和拓扑学。其目的是提高我们对辛世界中最近出现的某些对象（如霍费尔度量）的理解，并将它们应用于解决一些几何和动力学性质的老问题。具体方向包括研究行动的高阶格表面上，以下齐默的计划，简单的某些群体的转变，并存在准态射。过去二十五年见证了诞生了一个显着的数学领域，即辛拓扑。该领域与纯数学和现代物理学的许多领域都有联系。一些历史发展与格罗莫夫的名字有关，他提出了伪全纯曲线的概念，以及霍费尔的名字，他在经典的哈密顿动力学领域引入了一个引人注目的新度量不变量。在这个项目中，PI计划使用这些现在已经不可避免的新对象和技术，与纯数学的其他领域建立新的联系，如几何群论。