My goal in this series of talks is to give a complete proof of a recent theorem by B-Fisher-Miller-Stover showing that an hyperbolic manifold which admits infinitely many maximal totally geodesic submanifolds is arithmetic. A main ingredient in this proof is the method of AREA, Algebraic Representations of Ergodic Actions, developed in recent years by B-Furman, which I will explain in details. Coupling this method with some results of Ratner's Theory gives a clear and conceptual proof, modulo one technical detail, going under the name "compatibility", which I will explain. After completing the proof I will discuss the analogue theorem for complex hyperbolic manifolds and indicate the difficulty of its proof, stemming from possible "incompatibility", and how to go around it using Hodge theory and results from incidence geometry.
Manchester Building (Hall 2), Hebrew University Jerusalem
Uri Bader (Weizmann)