Limit distributions on homogeneous spaces: Interplay of dynamics and number theory

在数论的许多问题中，人们经常认识到群作用下的对称性和不变性；通过对李群齐次空间上子群作用动力学的理解，可以对这类问题有更深刻的认识。在这种情况下，有必要描述齐次空间子流形上光滑测度的平移序列的极限分布。在他早期对一些典型案例的研究的基础上，首席研究员在这个项目中寻求开发同质动力学的新技术，以提供流形和翻译元素的非常一般的自然条件，从而导致极限测度的平均分布。这项工作将有助于解决丢芬图近似的度量性质、群变异上的积分点和有理点计数以及与某些几何构型相关的几何有限群的轨道上的点计数等问题。从更广泛的角度来看，本研究将应用遍历理论和动力系统的概念和方法，研究流动中一般粒子的长期行为，这本来是一个物理问题，以回答关于整数的深层性质的问题。这两个截然不同的学科之间的概念桥梁使用并影响了其他领域，如表示理论、分析和微分几何，更不用说物理学了。

In a wide variety of questions in number theory one often recognizes symmetries and invariance under group actions; and deep insights can be gained into such problems by understanding dynamics of subgroup actions on homogeneous spaces of Lie groups. In several such instances it is necessary to describe limiting distributions of sequences of translates of smooth measures on submanifolds of homogeneous spaces. Building up further from his earlier work on some typical cases, the principal investigator seeks in this project to develop new techniques in homogeneous dynamics to provide very general natural conditions on the manifolds and translating elements that lead to equidistribution of the limit measures. The work should lead to resolutions of some questions on metric properties of Diophantine approximation, counting integral and rational points on group varieties, and counting points on orbits of geometrically finite groups associated with certain geometric configurations. From a broader perspective, the research will apply the concepts and methods of ergodic theory and dynamical systems about the long-time behavior of a generic particle in a flow, which is originally a physical problem, to answer questions about deep properties of whole numbers. This conceptual bridge between two very different disciplines uses and influences other areas like representation theory, analysis, and differential geometry, not to mention physics.