Equidistribution on homogeneous spaces for orbits of discrete groups beyond lattices

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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.