Rich representations and superrigidity

This talk will be about joint work with Baldi, Miller, and Ullmo that uses dynamics and/or Hodge theory to study rigidity problems for representations of real and complex hyperbolic lattices, especially those with many properly immersed totally geodesic subspaces (which are among the most well-studied lattices). Very roughly, rich representations are those for which the image of the representation has an action that respects this abundant collection of subgroups in some way, for instance the homomorphism induced by a map f : M -> N where infinitely many properly immersed totally geodesic subspaces of M map into a properly immersed totally geodesic subspace of N. The geometric motivation for defining rich representations has appeared in previous work of several people in several contexts. I will describe settings where we can show that rich representations are superrigid, including some progress toward a question of Siu about whether holomorphic embeddings between higher-dimensional complex hyperbolic manifolds must be totally geodesic.