The Drury-Arveson space was introduced by Drury in 1978, and it the years it proved to be the right analog of the Hardy space in several complex variables. It is a reproducing kernel space of holomorphic functions in several complex variables, and it lies at a very interesting threshold from the potential theoretic point of view. The plan is giving an introduction to the theory of the Drury-Arveson space, touching on a selection of topics ranging from the origins of the theory to very recent results. Here is a list of the topics I would like to cover.
(i) The Drury-Arveson space, its multiplier space, and von Neumann's inequality for commuting rows of operators.
(ii) The Complete Nevanlinna Property and the universal property of the Drury-Arveson space in connection to it.
(iii) A sub-Riemannian analysis of the reproducing kernel.
(iv) Carleson measures.
(v) Interpolating sequences.