James's Den

Mathematics

Mathematics

I've retired from mathematics, but I did my PhD at Newcastle on functional analysis, in particular in operator theory. Most of my research focussed on applying operator theoretical techniques to complex analysis problems, and vice versa, using Hardy spaces

I'll keep all my old papers up here though, so you'll always be able to get hold of my work.

Mathematical Papers

Some Applications of the Theory of Test Function Realisations

This is my PhD thesis. It's mostly taken from the papers Counterexamples to Rational Dilation on Multiply Connected Domains and Test Functions in Constrained Interpolation. However, some previously omitted proofs are included, and there is more background, and discussion of possible future work, so it may be more appropriate for researchers who are new to the field.

Abstract

We give two applications of the theory of test function realisations. These were developed in connection with interpolation problems. In the first, we use test function methods to construct a counterexample to the rational dilation conjecture. That is, we give a class of spaces and operators on them, such that the operators do not have rational boundary dilations. The second is a test function realisation for the algebra of functions on the disc with zero derivative at the origin. The methods used in this second application could also be used to find test function realisations for other algebras, or in interpolation problems.

Test Functions in Constrained Interpolation

Abstract

We give a set of test functions for H 1 ∞ , an algebra of holomorphic functions with constraints on derivatives, whose interpolation problem was studied by Davidson, Paulsen, Raghupathi and Singh. We show that this set of test functions is minimal, by relating these ideas to realisation and interpolation problems.

A version of this paper is also available on ArXiv.

Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains

Abstract

We show that if R is a compact domain in the complex plane with two or more holes and an anticonformal involution onto itself (or equivalently a hyperelliptic Schottky double), then there is an operator T which has R as a spectral set, but does not dilate to a normal operator with spectrum on the boundary of R.

A version of this paper is published in Complex Analysis and Operator Theory. The original publication is available at www.springerlink.com

Presentations

Examples and Applications of Test Functions Berlin, Germany, Thursday 15th July 2010

I gave this talk at IWOTA 2010, to an audience of operator theorists. I introduce the idea of a collection of test functions, and give some examples of test functions (including new, finitely generated sets), and discuss what can and can't be done with these examples. The talk includes a handout, with some useful theorems.

Please note that due to licensing restrictions on custom style files, this work contains material released under the GPL.

Test Function Realisations and Agler-Herglotz Representations Guanajuato, Mexico, Tuesday 22nd September 2009; Bengaluru, India, Monday 19th April 2010

I gave this talk at IWOTA 2009, to an audience of operator theorists. I introduce the idea of a test function realisation, and describe the Agler-Herglotz representation technique, which has been used by various authors to find test function realisations

I also gave a slightly extended version of this talk at a seminar in Bengaluru, Karnataka, India.

Please note that due to licensing restrictions on custom style files, this work contains material released under the GPL.

Test Functions and Constrained Interpolation Newcastle, Thursday 11th December 2008 and Leeds, Wednesday 17th December 2008

I gave this talk at an internal analysis seminar at Newcastle University, and a student conference on analysis in Leeds. It discusses Nevanlinna-Pick type problems, the test function approach, and my work on applying test functions to constrained interpolation problems.

Please note that due to licensing restrictions on custom style files, this work contains material released under the GPL.

The Axiom of Choice and its consequences Newcastle, Thursday 8th February 2007

I gave this talk at a general-audience postgraduate seminar in Newcastle. It gives a brief introduction the the Axiom of Choice.

Nevanlinna-Pick Interpolation Durham, Wednesday 21st May 2008

I gave this talk at a general-audience postgraduate seminar in Durham. It covers the Nevanlinna-Pick interpolation problem, its applications, and the techniques involved in working with Nevanlinna-Pick type problems.

Please note that due to licensing restrictions on third party content, this work is licensed under the GFDL.

Rational Dilation Newcastle, Tuesday 27th November 2007; Newcastle, Wednesday 13th January 2010; Munnar, India, Saturday 10th April 2010

I gave this talk at an internal analysis seminar at Newcastle University. It discusses the rational dilation problem, giving an introduction and summary of current results, and a discussion of a recent result of mine in the field.

I also gave a version of this talk at the 2010 Young Functional Analysts Workshop, in Newcastle. This was to a slightly more general audience, so took a less rigourous approach. I also gave this version of this talk at a small workshop in Munnar, Kerala, India.

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