The Dirichlet Space on the bi-disc

The analytic Dirichlet space on the complex disc is rather well understood and its counterpart on the unit ball in several complex dimensions shares with it many important features. For instance, there is a natural underlying (sub-Riemannian) geometry, with the obvious one-parameter family of metric balls having center at a given point. Things become drastically different in the bidisc, where genuine two-parameter phenomena appear. I will review what is known about this holomorphic space, including recent results in collaboration with Pavel Mozolyako, Karl-Mikael Perfekt, Giulia Sarfatti.