The problems to be investigated are in the area of dynamical systems on homogeneous spaces which are obtained as a quotient of a Lie group by a discrete subgroup. Such spaces arise naturally from problems in number theory. The main objective is to continue the path of establishing dynamics on homogeneous spaces as a powerful tool to approach certain problems in number theory. The following problems will be main objects of the study: (i) "effectivization" of orbit closure and equdistribution results for orbits of certain subgroups (ii) establishing "rigidity" results for invariant measures and orbits of ''certain subgroups" on homogeneous spaces; special attention will be given to the case of Radon measures on geometrically finite manifolds and Raghunathan's conjecture in positive characteristic setting.
Lie groups and their discrete subgroups are one of the central objects of study in mathematics. During the past 40 years, techniques and ideas from dynamics on homogeneous spaces have been fruitful in answering certain long standing questions in number theory, in particular in Diophantine approximation. The relevant results are mostly in the rigidity theory of the kind that a rather weak initial information about an object almost fully classifies the object. This proposal seeks continuation of this path.
