Equidistribution on homogeneous spaces for orbits of discrete groups beyond lattices

这个建议提出研究的计数和均匀分布问题的几何和算术对象的齐次空间所产生的轨道离散子群的李群不一定是格。指导方针是使用技术从各种不同的领域，如调和分析，动力学的群体行动，双曲几何，遍历理论，自守形式的半单代数群，以描述渐近分布的一个无限序列的子流形所产生的数论和几何的情况。然而，这些技术大多是发达国家只为研究轨道的晶格和我们的主要重点在于调查在何种程度上，我们可以进一步发展类似的技术轨道的离散子群与无限covolume.The拟议的项目在于交叉的几个领域的数学，包括李群，解析数论，双曲几何，克莱因集团，遍历理论和动力学。它涉及几个研究领域之间的联系，并对不同数学领域的各种主题有着深刻的应用。

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.