Homogeneous Dynamics and Number Theory

本文研究的问题是齐次空间上的动力系统，它是由一个离散子群作为李群的商得到的。这样的空间自然产生于数论中的问题。主要目标是继续在齐次空间上建立动力学的路径，作为解决数论中某些问题的有力工具。以下问题将是研究的主要对象：(i)某些子群轨道闭合和均匀分布结果的“有效性”；（ii）建立齐次空间上不变测度和“某些子群”轨道的“刚性”结果；特别注意几何有限流形上的Radon测度和正特征设置下的Raghunathan猜想。李群及其离散子群是数学研究的中心问题之一。在过去的40年里，齐次空间动力学的技术和思想在回答数论中一些长期存在的问题上取得了丰硕的成果，特别是在丢番图近似中。相关的结果大多在刚性理论中，即关于一个物体的相当弱的初始信息几乎可以完全分类该物体。这项建议寻求继续这条道路。

The problems to be investigated are in the area of dynamical systems on homogeneous spaces which are obtained as a quotient of a Lie group by a discrete subgroup. Such spaces arise naturally from problems in number theory. The main objective is to continue the path of establishing dynamics on homogeneous spaces as a powerful tool to approach certain problems in number theory. The following problems will be main objects of the study: (i) "effectivization" of orbit closure and equdistribution results for orbits of certain subgroups (ii) establishing "rigidity" results for invariant measures and orbits of ''certain subgroups" on homogeneous spaces; special attention will be given to the case of Radon measures on geometrically finite manifolds and Raghunathan's conjecture in positive characteristic setting. Lie groups and their discrete subgroups are one of the central objects of study in mathematics. During the past 40 years, techniques and ideas from dynamics on homogeneous spaces have been fruitful in answering certain long standing questions in number theory, in particular in Diophantine approximation. The relevant results are mostly in the rigidity theory of the kind that a rather weak initial information about an object almost fully classifies the object. This proposal seeks continuation of this path.