Program

The program is still to be confirmed, but will consist of two-and-a-half days of postgraduate and invited talks, with poster presentations another option.

This year we will have invited talks by established Mathematicians in this field.

For a more detailed program and list of talks, please see the attached PDF.

Abstracts

Apologies for the raw LaTeX markup in some abstracts; please see the PDF for a formatted version.

Invited Talks

Michael Dritschel (Newcastle)

A Tour of von Neumann's Inequality

In 1951 John von Neumann published the following inequality which bears his name:

If $p$ is a complex polynomial and $T$ is a bounded Hilbert space operator with $\|T\| \leq 1$, then $\|p(T)\| \leq \|p\|_\infty$, the supremum norm of $p$ over the unit disk.

The inequality has immediate application to functional calculus questions, and for this reason (among others) it is interesting to know for which function algebras and operators inequalities of this sort are attained.

For example, given a function algebra $\mathcal A$ over a complex domain with norm denoted by $\|\cdot\|_\infty$ (though perhaps not the supremum norm!), one could look for conditions on an operator $T$ such that $f(T)$ makes sense for $f\in A$ and $\|f(T)\| \leq \|f\|_\infty$.

Or one might also ask: given a collection of operators, are there natural domains and function algebras over these domains such

that some version of von Neumann's inequality holds?

What about algebras over $\mathbb C^n$ or tuples of operators (commuting or not)?

We will touch on many of these problems, discuss some of the recent work in this area, list a few of the many applications and point to open problems around this fascinating inequality.

Simon Eveson (York)

The Strange Ubiquity of Rank 1

The simplest linear operators on a Banach space are those of rank 1. This talk describes three different situations in which relatively complicated linear systems exhibit very simple rank 1 behaviour in an asymptotic sense. This leads to a question (to which I don't know the answer!): are these isolated examples, or part of some more general theory?

Steve Power (Lancaster)

Crystal Frameworks and Rigidity Operators

A finite bar-joint framework in three dimensions has an associated rigidity matrix which detects infinitesimal rigidity and flexibility of the framework.

Infinite bar-joint (bond-node) frameworks, with periodic structure, play a role in mathematical models for Rigid Unit mode vibrations (low energy phonons) in material crystals.

I shall introduce their infinite rigidity matrices and rigidity operators and present some of the foundations of a fledgling topic.

In particular I hope to indicate,

1. how one may compute the rigid unit mode spectrum of a crystal framework, and
2. how one might develop a Hilbert space theory for "square summable flexes".