Exploring realization theorems and some of their recent generalisations, as well as their application

Given a normed function algebra, how do we describe those elements which are in the unit ball of the algebra?

There turn out to be a number of ways (collected under the umbrella of a realization theorem), and the various descriptions allow us to explore important properties of the algebra.

What we discover is that there is a subtle interplay between such algebras and operator theory - operator theory yields information about the algebra, while the algebras associated to operators can help to untangle their structure.

Though this will be a pure mathematics project, it is worth mentioning that realizations are also important in engineering applications - indeed the transfer function representation of an element of the unit ball of the algebra which forms a key parts of a realization theorem initially came from engineers studying feedback control.

This project will explore realization theorems and some of their recent generalisations, as well as their application. In particular, we’ll consider in this way rational dilation problems, which attempt to gather information about an operator by viewing it as a piece of some simpler (in this case, normal) operator.

It should be noted, that successful applicants will also be given the opportunity to complete teaching and demonstrating duties within the school amounting to up to £1500 per annum.