Equidistribution on homogeneous spaces for orbits of discrete groups beyond lattices

这项建议建议研究齐次空间上几何对象和算术对象的计数和等分布问题，这些对象作为李群的离散子群的轨道出现，而李群不一定是格。其指导思想是利用调和分析、群作用动力学、双曲几何、遍历理论和半单代数群的自同构形等不同领域的技巧来描述在数论和几何情形中出现的无限序列的子流形的渐近分布。然而，这些技术大多只用于格点轨道的研究，我们的主要重点是研究在多大程度上可以进一步发展类似的技术来研究具有无限余体积的离散子群的轨道。所提出的项目涉及李群、解析数论、双曲几何、Kleinan群、遍历理论和动力学等多个数学领域。它涉及几个研究领域之间的联系，并在不同数学领域的不同主题中有深入的应用。

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.