Ratner's project will 10 extend Raghunathan's conjectures to p-adic Lie groups, 2) study the dynamical properties of the geodesics and the horocycle flows on homogeneous spaces of SL(2,Q_p), 3) apply previous results on orbit closures of unipotent flows to find asymptotic growth rate of integral solutions of quadratic inequalities, and 4) study Principal Congruence groups and the Ramanujan-Selberg Conjecture. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.
