A non-commutative Julia Inequality

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there

Algebraic pairs of isometries

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial

Aspects of Non-commutative Function Theory

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable

Cusp Algebras

We consider simple cusp algebras, that is certain subalgebras of the algebra of holomorphic functions on a disk that are annihilated by some distributions living on a singleton. We determine when these algebras can be holized in two dimensions, and when these holizations are globally biholomorphic