Arithmetic, Geometric, and Ergodic Aspects of the Theory of Lie Groups and Their Discrete Subgroups

李群及其离散子群理论的算术、几何和遍历方面

• 批准号: 9800607
• 负责人: Gregory Margulis
• 金额: $ 55.98万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800607&HistoricalAwards=false
• 关键词: Arithmetic; Geometric; Ergodic; Aspects; Theory

AbstractMargulisThe problems addressed in this research proposal are in the area of the theoryof Lie groups and their discrete subgroups. The common theme is to study thistheory in connection with number theory, geometry and ergodic theory. One ofthe main objectives is to continue the program of establishing a homogeneousspace approach as a powerful tool in number theory and, in particular, in thetheory of Diophantine approximations. It is also proposed to study a variety ofquestions about actions of Lie groups and their discrete subgroups onhomogeneous spaces, manifolds and general metric spaces. In particular theprincipal investigator plans to continue his joint work with othermathematicians on properly discontinuous groups of affine transformations and onlocal rigidity for standard algebraic actions. The technique of Lie grouptheory, reduction theory, ergodic theory and the theory of dynamical systemswill be used throughout the proposed work.The theory of Lie groups and their discrete subgroups is one of the centralfields in mathematics. Historically this theory was mostly motivated by itsconnections with geometry, mechanics and the theory of differential equations. During the last 15-20 years it was realized that some aspects of the theory canbe applied to solve certain problems in number theory and related topics, whichcould not be tackled by other methods for several decades. It is suggested inthe proposal to continue this development in order to obtain new applications innumber theory, dynamical systems and geometry. In general, the proposal shouldestablish new connections between discrete and continuous in mathematics.

摘要margulis在这项研究建议中解决的问题是谎言组理论及其离散亚组的领域。 共同的主题是研究与数字理论，几何学和千古理论有关的理论。 主要目标之一是继续将同质空间方法建立为数字理论的强大工具，尤其是在Diophantine近似值中的计划。 还建议研究有关谎言群体及其离散亚组的各种问题，均质空间，流形和一般度量空间。 特别是，原理调查员计划继续与其他有关的人共同合作，以适当不连续的仿射转变和本地僵化群，以实现标准代数行动。 在整个拟议的工作中，都可以使用谎言群体理论，还原理论，千古理论和动力系统设备理论的技术。谎言群体及其离散亚组的理论是数学中的中央场之一。 从历史上看，这一理论主要是由它与几何学，力学和微分方程理论的联系所激发的。在过去的15 - 20年中，人们意识到该理论的某些方面可以应用于数字理论和相关主题中的某些问题，几十年来其他方法不得解决。 建议继续进行这种发展，以获取新的应用理论，动力学系统和几何形状。 通常，该提案应在数学中建立离散与连续之间的新联系。