Equidistribution on homogeneous spaces for orbits of discrete groups beyond lattices

本文研究了几何和算术对象在齐次空间上作为李群的离散子群轨道产生的计数和均分问题，这些子群不一定是格。本文的指导思想是利用调和分析、群作用动力学、双曲几何、遍历理论和半简单代数群的自同构形式等不同领域的技术来描述数论和几何情况下产生的无限子流形序列的渐近分布。然而，这些技术大多只适用于晶格轨道的研究，我们的主要重点是研究在多大程度上我们可以进一步发展类似的技术来研究具有无限共体积的离散子群的轨道。提出的项目涉及几个数学领域的交叉，包括李群、解析数论、双曲几何、Kleinian群、遍历理论和动力学。它涉及到几个研究领域之间的联系，并在数学的不同领域的各种主题有深入的应用。

This proposal proposes to study the counting and equidistribution problems of geometric and arithmetic objects on homogeneous spaces arising as orbits of discrete subgroups of Lie groups which are not necessarily lattices. The guide line is to use the techniques from various different fields such as harmonic analysis, dynamics of group actions, hyperbolic geometry, ergodic theory, and automorphic forms of semisimple algebraic groups in order to describe the asymptotic distribution of an infinite sequence of submanifolds arising in number theoretic and geometric situations. However most of these techniques are well developed only for the study of orbits of lattices and our main focus lies in investigating to what extent we can further develop similar techniques for orbits of discrete subgroups with infinite co-volume.The proposed projects lie in the intersection of several fields of mathematics including Lie groups, analytic number theory, hyperbolic geometry, Kleinian groups, ergodic theory and dynamics. It involves connections between several areas of research, and has deep applications to various topics in different areas of mathematics.