Matthew Stover

A very old, basic problem is classifying closed n-manifolds admitting a
metric of constant (holomorphic) sectional curvature. The most
mysterious case is constant curvature -1, that is, (complex) hyperbolic
manifolds; these divide further into "arithmetic" and "nonarithmetic"
manifolds. It is not at all evident from the definitions that this
distinction has anything to do with the differential geometry of the
manifold. Uri Bader, David Fisher, Nicholas Miller and I gave a
geometric characterization of arithmeticity in terms of properly
immersed totally geodesic submanifolds, answering a question due
independently to Alan Reid and Curtis McMullen. I will describe how
(non)arithmeticity and totally geodesic submanifolds are connected, then
how this allows us to import tools from ergodic theory and homogeneous
dynamics originating in groundbreaking work of Margulis to prove our
characterization.