In this second (expository) seminar we see that the unit ball in several complex variables carries a Riemannian metric which is invariant under automorphisms. Its restrictions to horocycles converge, after rescaling, to a sub-Riemannian metric on the boundary, which has a natural Lie group (Heisenberg) structure. These same metrics also arise from the study of the Drury-Arveson space on the unit ball (a Hilbert space modelling contractive tuples of communitng operators on Hilbert spaces), as well as from problems in control theory.