## Mathematical Work

Until 2015, I worked as an academic mathematician at the University of Leeds.

• My research interests started out in Functional Analysis, specifically Banach Spaces, but evolved towards algebra, working in Banach Algebra theory, and then extensively with the Operator Algebraic approach to Quantum Groups. I was an editorial advisor for the Glasgow Mathematical Journal, and for the four London Mathematical Society journals.
• I won an EPSRC research grant, which employed a postdoc.
• I taught a wide range of courses, from a Master's Course with 2 students, through to 1st year modules with over 150 students.
• I spent nearly four years as the admissions tutor, recruiting 250+ single and joint honours students to a variety of Mathematics courses, managing day to day administration, academic issues, and marketing work.

Downloadables below include lecture notes for Linear Analysis I, resources from outreach talks, some informal unpublished notes, Galois Theory lecture notes from Cambridge, and a couple of half-finished projects.

## Publications

1. "The approximation property implies that convolvers are pseudo-measures" with Nico Spronk. To appear in Studia Math. This paper is available on the ArXiV.
2. "Around Property (T) for quantum groups" with Adam Skalski and Ami Viselter. To appear in Communications in Mathematical Physics. This paper is available on the ArXiV.
3. "Categorical aspects of quantum groups: multipliers and intrinsic groups". Published in the Canadian Journal of Mathematics volume 68 (2016), 309--333. This paper is available on the ArXiV.
4. "Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras". Published in J. Funct. Anal. Volume 269 (2015) 683--704. This paper is available on the ArXiV.
5. "Quantum Eberlein compactifications and invariant means" with Biswarup Das. To appear in the Indiana University Mathematics Journal Vol 65 No 1 (2016) 307--352. This paper is available on the ArXiV.
6. "Remarks on the Quantum Bohr Compactification". Published in Illinois J. Math. Volume 57, Issue 4, 1131--1171. This paper is available on the ArXiV.
7. "The Haagerup property for locally compact quantum groups" with Pierre Fima, Adam Skalski and Stuart White. Published in Crelle's Journal (Journal für die reine und angewandte Mathematik) Volume 2016 Issue 711, 189--229. This paper is available on the ArXiV.
8. "Equivalence of multi-norms" with H. Garth Dales, Hung Le Pham and Paul Ramsden. Published in Dissertationes Math. Volume 498 (2014) 53pp.
9. "Completely positive definite functions and Bochner's theorem for locally compact quantum groups" with Pekka Salmi. Published in J. Funct. Anal. Volume 264, Issue 7, 1525-1546. This paper is available on the ArXiV.
10. "Closed quantum subgroups of locally compact quantum groups" with Paweł Kasprzak, Adam Skalski and Piotr Sołtan. Published in Advances in Mathematics Volume 231 (2012) 3473-3501. This paper is available on the ArXiV.
11. "Completely positive multipliers of quantum groups". Published in International Journal Mathematics volume 23, no 12 (2012). This paper is available on the ArXiV.
12. "Completely bounded representations of convolution algebras of locally compact quantum groups" with Michael Brannan and Ebrahim Samei. Published in Münster Journal of Mathematics volume 6 (2013) 445-482. This paper is available on the ArXiV.
13. "Isometries between quantum convolution algebras" with Hung Le Pham. Published in The Quarterly Journal of Mathematics 64 (2013) 373-396. This paper is available on the ArXiV.
14. "Shift invariant preduals of $\ell_1(\Z)$" with Richard Haydon, Thomas Schlumprecht and Stuart White. Published in Israel J. Math. 192 (2012), 541–585. This paper is available on the ArXiV.
15. "Multi-norms and the injectivity of $L^p(G)$" with H. Garth Dales, Hung Le Pham and Paul Ramsden. Published in J. Lond. Math. Soc. Volume 86 (2012) 779-809. This paper is available on the ArXiV.
16. "Multipliers of locally compact quantum groups via Hilbert C$^*$-modules". Published in J. Lond. Math. Soc. Volume 84 (2011) 385-407. This paper is available on the ArXiV.
17. "Multipliers, Self-Induced and Dual Banach Algebras", published in Dissertationes Mathematicae volume 470 2010, 62pp. This paper is available on the ArXiV.
18. "A bicommutant theorem for dual Banach algebras". Published in the Mathematical Proceedings of the Royal Irish Academy 111 (2011) 21-28. This paper is available on the ArXiV.
19. "Characterising weakly almost periodic functionals on the measure algebra". This paper is available on the ArXiV. published in Studia Mathematica Volume 204, 2011, 213--234.
20. "Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces", published in Canadian Journal of Mathematics Volume 63, 2011, 798--825. This paper is available on the ArXiV.
21. "Operator Biprojectivity of Compact Quantum Groups", published in Proceedings of the American Mathematical Society Volume 138, 2010, 1349-1359. This paper is available on the ArXiV.
22. "Preduals of semigroup algebras" with Hung Le Pham and Stuart White, published in Semigroup Forum Volume 80, 2010, 61--78. This paper is available on the ArXiV.
23. "Reiter's properties (P_1) and (P_2) for locally compact quantum groups", with Volker Runde, published in Journal of Mathematical Analysis and Applications Volume 364, Issue 2, 2010, 352-365. This paper is available on the ArXiV. See Corrections!.
24. "Weakly almost periodic functionals on the measure algebra", published in Mathematische Zeitschrift Volume 265 (2010) 285-296. This paper is available on the ArXiV.
25. "Conditions implying the uniqueness of the weak$^*$-topology on certain group algebras" with Hung Le Pham and Stuart White. Published in Houston Journal of Mathematics Volume 35 (number 1) 2009, 253-276. This paper is available on the ArXiV.
26. "Can B(l^p) ever be amenable?" with Volker Runde, published in Studia Mathematica Volume 188 (number 2) 2008, 151-174. See also Erratum, Studia Mathematica, Volume 195 2009, 297-298. This paper is available on the ArXiV. See Corrections!.
27. "Ultrapowers of Banach algebras and modules", published in Glasgow Mathematics Journal, Volume 50, Issue 03, 2008, 539-555. DOI:10.1017/S0017089508004400. This paper is available on the ArXiV.
28. "Amenability of ultrapowers of Banach algebras", published in Proceedings of the Edinburgh Mathematical Society, Volume 52, 2009, 307-338. DOI:10.1017/S0013091507001083. This paper is available on the ArXiV. See Corrections!.
29. "p-Operator Spaces and Figa-Talamanca-Herz Algebras", published in The Journal of Operator Theory, Volume 63, 2010, 47-83. This paper is available on the ArXiV.
30. "Arens-regularity of Algebras arising from Tensor Norms", published in the New York Journal of Mathematics Volume 13 (2007) 215-270. This is a free to access online journal.
31. "Dual Banach algebras: representations and injectivity", published in Studia Mathematica Volume 178 (number 3) 2007, 231-275. This paper is available on the ArXiV.
32. "Connes-amenability of bidual algebras", published in Mathematica Scandinavica Volume 99, Issue 2, 2006, 217-246. This paper is available on the ArXiV (warning: old version).
33. "Closed ideals in the Banach algebra of operators on classical nonseparable spaces" published in Math. Proc. Camb. Phil. Soc. Volume 140, Issue 02, March 2006, pp 317-332. DOI:10.1017/S0305004105009102. Download preprint: Acrobat PDF.
34. "Semi-simplicity of B(E)**" with Charles Read, published in Journal of Functional Analysis, Volume 219, Part 1, (2005), pp 177-204. DOI:10.1016/j.jfa.2004.04.014. Download preprint: Acrobat PDF.
35. "Arens regularity of the algebra of operators on a Banach space" published in Bulletin of the London Mathematical Society, Volume 36, Part 4, (2004), pp 493-503. DOI:10.1112/S0024609303003072. Download preprint: Acrobat PDF.

Some of my early papers have corrections, which can be download: corrections [PDF]. It seems I became better at checking as time went on!

## Teaching biography

#### Year 2014/15

I continue to be the admissions tutor. I am tutoring a group for MATH1025 Number Systems and for MATH1225 Introduction to Geometry.

#### Year 2013/14

I was now the (full-time) admissions tutor, and so have a much reduced teaching load. I tutored a group for MATH1025 Number Systems.

In the 2nd semester, I had a project student on MATH3083, and a number of students working on essays for MATH3000.

#### Year 2012/13

I continued to be the deputy admissions tutor, working with Dr Daniel Read. I gave the 2nd half of the course MATH1010 Mathematics I, this time with a slightly re-written linear algebra section, and worked with Dr Phil Walker, pioneering e-assessment on this module. I tutored a student on the project module MATH5004 and run tutorials for MATH1010 and MATH1225.

In the second semester, I ran workshops for the new module MATH2016 Analysis, tutored on MATH3000, and continued tutoring on MATH5004.

#### Year 2011/12

I lectured the 2nd half of the course MATH1010 Mathematics I, a new course--my part featured partial differentiation and an introduction to matrix algebra-- and also also tutored students for this module. This module was introduced after a major re-working of 1st and 2nd year modules. I also tutored students on the project module MATH3083, and ran workshops for MATH2015, Analysis 2 (the final time this module ran), tutored students for MATH1060 and finally helped out with MATH3000 again.

I also ran a reading course MATH5002 based around the book "Introduction to Banach Spaces and Algebras" by Allan. Please see below for some teaching resources associated to this.

I also began my new job as the deputy admissions tutor, working with Dr Daniel Read. Not strictly teaching, but rather recruiting the next generation of undergraduates.

#### Year 2010/11

I lectured the first semester of MATH1035, Analysis I (which is the final year this course is running) and also the course MATH2200, Linear Algebra 2, again a "problematic" course for which I rewrote much of the lectures notes and support material. As ever, I also supervised students on the project modules MATH3000 and MATH3083, and tutored on MATH1060.

#### Year 2009/10

I again lectured MATH5015, Linear Analysis I, and MATH3181, Inner-products and Metric spaces in the second semester. Additionally, in the first semester, I took over the course MATH1035, Analysis I, which was proving problematic. I rewrote the lecture notes and example sheets, merging them (modelling on the Warwick Analysis course) to introduce more coupling between lectures and "homework", in an abstract course that requires a high degree of active learning. This was a case-study at our "teaching away day".

#### Year 2008/09

I again lectured MATH5015, Linear Analysis I, in the first semester, and MATH3181, Inner-products and Metric spaces in the second semester. This was an existing course, and as part of my teaching award, I rewrote the example sheets (to introduce more self-discovery into the module) and produced a number of videos, to supplement lectures.

#### Year 2007/08

I lectured MATH5015, Linear Analysis I, in the first semester. This was a new course, and I wrote from scratch lecture notes and supporting exercises.

## Introduction to Banach Spaces and Algebras

In 2012 I lead a reading course on the book "Introduction to Banach Spaces and Algebras" by Graham Allan (an old lecturer of mine). See Amazon link or OUP link to buy the book. Here are various teaching materials, in LaTeX and PDF formats.

## Popular talks and outreach

I helped Richard Elwes give a popular talk, to sixth-form students, on knot theory. This was part of the Leeds Festival of Science. Special thanks to Ruth Holland, Hazel Kendrick and David Pauksztello. We repeated the effort as a "Reach for Excellence session".

The following are some handouts (with LaTeX source) which I produced. The margins of the PDF files are off, probably because I used pstricks, and hence ps2pdf, instead of pdflatex.

Recently, again with Richard Elwes, I gave a session as part of the Leeds festival of science on the subject of Pick's Theorem. It is always tricky leading A-Level students through a pure mathematics proof, but it's also very pleasing to see some students (often not those you expect!) suddenly "get" the point, and start to really understand something new.

I run a "Quiz night" as part of the annual 6th Form Conference held at the university. Thanks for Alan Slomson, on whose idea this was based. Contact me if you would like further information: I won't post the quiz, to discourage cheating!

## Talks

Most of my research talks, which may still be of interest.

## Notes, Writing

### MathOverflow

I used to spend some time on MathOverflow but not so much now.

### MathSciNet

I wrote exactly 60 MathSciNet reviews: if you have access, you can see the processed reviews here.

### Thesis

My thesis was mined fairly successfully for papers (I think 6 in the end) but the original may still be of interest:

### Informal and Unpublished Notes

• The group algebra derivation problem [PDF]: A self-contained proof of the group derivation problem, using the new proof by Bader et al.
• "Functorial properties of weakly almost periodic functionals on the measure algebra". This has been superseded by a later paper, as the referee noticed a much, much better way of proving the main result. However, perhaps the techniques used in this paper might be of interest: in particular, is there a non-commutative generalisation? This paper is available on the ArXiV, and will not otherwise be published (be aware that material in the first and final section is dealt with better in arXiv:0904.0436 (current version)).
• "Weakly almost periodic functionals, representations, and operator spaces": This paper shows that a completely contractive dual Banach algebra is isometric to a weak*-closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result has been shown independently by F. Uygul (math.FA/0702200 on the arXiv), using rather similar arguments to my presentation, which will hence remain only on the arXiv.
• Bases in Banach spaces [PDF]: Introduction to Schauder bases, basic sequences, unconditional bases, and the Gowers dichotomy theorem. Prepared for a talk.
• Interpolation spaces [PDF]: Some notes on Beauzamy's book. Prepared for a talk.

### Galois Theory

These are, verbatim, the notes presented by Dr. P.M.H. Wilson in Michaelmas term, 1999. I LaTeXed them, added a few notes of explanation (to account for my lack of understanding of the course Groups, Rings and Fields). Briefly, the notes cover field extensions, separability, algebraic closures, normal extensions and Galois extensions, Galois groups of finite fields, cyclotomic extensions, Kummar theory, solving by radicals and the insolubility of the general quintic by radicals.

• Local version in Acrobat (395Kb) format.
• Latex source available upon request.

### Unfinished projects

These are two long term projects which never really got finished. I'm providing LaTeX sources in case anyone finds a use. Licensed under the Creative Commons Attribution 3.0 Unported License.

#### Compact quantum groups

This started as some private notes to fix notation, but evolved into the beginnings of a set of lecture notes, or a book. In particular, there is a self-contained (and hopefully not too bug ridden) treatment of Woronowicz's axioms through to developing corepresentation theory (similar to the presentation of Maes--Van Daele, but I work with left multiplicative unitaries, to align better with the theory of Locally Compact Quantum Groups).

#### Unitaires multiplicatifs

This is an unfinished attempt to both translate the classic Baaj--Skandalis paper into English, and at the same time to update the paper (for example, it was written contemporaneously with Woronowicz's "Compact Quantum Groups" paper and so only treats compact matrix quantum groups). Very unfinished.