Georgia Institute of Technology College of Sciences

In the 1990's Marina Ratner published a famous series of papers showing that ergodic measures invariant under unipotent flows over quotients $\Gamma/G$ are homogeneous. From this, she deduced many other remarkable properties for these flows (e.g that the closure of orbits are homogeneous and that orbits are uniformly distributed in their closures). To prove this result will require several lectures, but already the case of horocycle flow in $\Gamma/SL(2, \mathbb{R})$ presents several or her ideas. In this talk we will present the ideas of the proof in this case and present an application due to Margulis.