I will discuss some recent work showing that finite volume real and complex hyperbolic manifolds having infinitely many maximal totally geodesic submanifolds are arithmetic. I will put this in the context of what we do and (mostly) don't know about real and complex hyperbolic manifolds, particularly in dimension at least 4. Key ingredients in the proofs depend on homogeneous dynamics and algebraic dynamics and it is tempting to believe these might be relevant to some remaining mysteries. This is based on joint work with Bader, Miller and Stover and also some earlier joint work with Lafont, Miller and Stover.

Zoom link: https://iu.zoom.us/j/661711533?pwd=RTFVTjMrQ1pYTCtIZzIvVGVvODV2QT09
password is 076877 if needed. Speaker(s): David Fisher (Indiana University)