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: 0801195
• 负责人: Gregory Margulis
• 金额: $ 81.02万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801195&HistoricalAwards=false
• 关键词: Arithmetic; Geometric; Ergodic; Aspects; Theory

The problems to be investigated are in the area of the theory of Lie groups and their discrete subgroups. One of the main objectives is to continue the program of establishing a homogeneous space approach as a powerful tool in number theory. Special attention will be given to the problem of "effectivization" in Oppenheim conjecture and its quantitative generalizations. It is also proposed to continue to study recurrence properties of random walks on Lie groups and their discrete subgroups on homogeneous spaces, manifolds and general metric spaces.The theory of Lie groups and their discrete subgroups is one of the central fields in mathematics. During the last few decades, it was realized that some aspects of the theory can be applied to solve certain problems in number theory and related topics, which could not be tackled by other methods. This proposal is related to rigidity theory that studies phenomena when rather weak data about geometric and mathematical objects determines completely or almost completely the structure of those objects.

要研究的问题属于李群及其离散子群的理论领域。 主要目标之一是继续建立齐质空间方法作为数论中的强大工具。 将特别关注奥本海姆猜想及其定量推广中的“有效化”问题。 并提出继续研究齐次空间、流形和广义度量空间上李群及其离散子群上的随机游走的递推性质。李群及其离散子群理论是数学的中心领域之一。 在过去的几十年中，人们意识到该理论的某些方面可以应用于解决数论和相关主题中的某些问题，而这些问题是其他方法无法解决的。 该提议与刚性理论有关，该理论研究当几何和数学对象的相当弱的数据完全或几乎完全决定这些对象的结构时的现象。