Ergodic Theory and Homogeneous Dynamics

The study of flows on homogeneous spaces $G/\Gamma$ — where $G$ is a Lie group and $\Gamma$ a lattice — has led to deep interactions between dynamics, geometry, and number theory. We focus on the action of unipotent one-parameter subgroups and present Ratner's measure classification theorem: every ergodic invariant probability measure for a unipotent flow on $G/\Gamma$ is algebraic. We discuss applications to the Oppenheim conjecture on values of quadratic forms and to counting lattice points \in regions of symmetric spaces.