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

【摘要】本文研究的是李群及其离散子群的理论问题。主要目标之一是继续建立齐次空间方法作为数论的有力工具的计划。本文将特别关注奥本海姆猜想中的“有效性”问题及其定量推广。并提出继续研究李群及其离散子群在齐次空间、流形和一般度量空间上的随机游动的递推性质。李群及其离散子群理论是数学研究的核心领域之一。在过去的几十年里，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，这些问题是其他方法无法解决的。这一建议与刚性理论有关，刚性理论研究的现象是，当几何和数学对象的相当弱的数据完全或几乎完全决定了这些对象的结构。

AbstractMargulisThe 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.