Multivariable Hardy spaces and the classification of universal dilation algebras

In this talk, we will discuss what is special about the Hardy spaces $H^2(\mathbb{D})$ and its
multiplier algebra $H^{\infty}(\mathbb{D})$, from the point of view of operators algebras and
function theory. I will present two generalizations of the pair $H^2$ and $H^{\infty}$ to the
multivariable setting. One commutative and one noncommutative. We will then discuss a natural
classification question that arises in the multivariable setups of algebras of analytic functions
on subvarieties of the unit ball. These algebras arise naturally as universal operator algebras
of a class of row contractions. Only basic familiarity with operators on Hilbert spaces and complex
analysis is assumed.