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

Non-commutative holomorphic functions on Operator domains

We characterize functions of $d$-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets

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

Toral Algebraic Sets and Function Theory on Polydisks

A toral algebraic set $A$ is an algebraic set in \C^n whose intersection with \T^n is sufficiently large to determine the holomorphic functions on $A$. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set

Induced Chern-Simons terms

We examine the claim that the effective action of four-dimensional SU(2)_L gauge theory at high and low temperature contains a three-dimensional Chern-Simons term with coefficient being the chemical potential for baryon number. We perform calculations in a two-dimensional toy model and find that the existence of the term is rather subtle.Comment: 12 pages, LaTe