The Drury-Arveson space $H^2_d$, also known as symmetric Fock space, is a Hilbert space of holomorphic functions on the Euclidean unit ball in $\mathbb C^d$.

Its origins can be traced back to at least the 1970s, but the subject really gained momentum around the turn of the millennium.

The Drury-Arveson space is now widely considered to be the 'right' generalization of the Hardy space on the disc to the unit ball for many purposes.

The importance of the Drury-Arveson space can be seen from two distinct appearances of this space.

(1) Multivariable operator theory: In the study of contractions on Hilbert space, the unilateral shift plays a key role.

A natural multivariable generalization of contractions are commuting row contractions. In this setting, the tuple $M_z = (M_{z_1},\ldots,M_{z_d})$ of operators of multiplication by the coordinate functions on $H^2_d$ plays the role of the unilateral shift. In particular, the natural analogues of von Neumann's inequality and of Sz.-Nagy's dilation theorem for commuting row contractions take place in the Drury-Arveson space.

(2) Complete Pick spaces: Complete Pick spaces are Hilbert function spaces satisfying a version of the Pick interpolation theorem. Important examples are the Hardy space and the Dirichlet space. The Drury-Arveson space is also a complete Pick space, but it is more than just an example.

A theorem of Agler and McCarthy shows that the Drury-Arveson space is a universal complete Pick space in the sense that every complete Pick space can essentially be identified with a restriction of the Drury-Arveson space.

In this mini course, I will provide an introduction to the Drury-Arveson space. Topics that will be covered include:

- different descriptions of the Drury-Arveson space

- row contractions, dilations and von Neumann's inequality

- pick spaces and universality of the Drury-Arveson space

- interpolating sequences