Event № 118
Suppose that one is given n points z_1, ... z_n in the unit disc and n complex numbers w_1, ..., w_n. It is always possible (and easy) to find an analytic function that interpolates this data, meaning that f(z_i) = w_i for all i. A more difficult problem is to determine whether or not there is an analytic function which interpolates the data and, in addition, is bounded on the disc by some given constant, say 1.
In 1916, G. Pick solved this problem, and gave an effective necessary and sufficient condition for the existence of such an interpolating function. Pick's theorem turns out to be best understood in the setting of Hilbert function spaces, and has been of great interest to mathematicians as well as engineers.
A couple of decades ago the question "for which algebras (other than the algebra of bounded analytic functions on the disc) does a theorem like Pick's theorem hold?" was raised, and eventually found a complete solution by Quiggin, McCullough and Agler & McCarthy. These algebras sometimes go under the name "Pick algebras". Subsequent work has clarified the structure of these algebras, and the classification of these algebras up to isomorphism has been one of my favourite problems in the last five years or so.
In my talk I will describe Pick's theorem, how it fits in the framework of Hilbert function spaces, what Pick algebras look like and what we know about the classification of these algebras.
The bottom line will be that every Pick algebra can be viewed as an algebra of bounded analytic functions on some analytic variety, and that the analytic varieties provide geometric invariants for the algebraic and operator algebraic structure of Pick algebras.