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Downloadables below include lecture notes for Linear Analysis I, resources from outreach talks, some informal unpublished notes, Galois Theory lecture notes from Cambridge, and a couple of half-finished projects.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

- "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.
- "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.
- "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.
- "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.
- "Remarks on the Quantum Bohr Compactification". Published in Illinois J. Math. Volume 57, Issue 4, 1131--1171. This paper is available on the ArXiV.
- "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.
- "Equivalence of multi-norms" with H. Garth Dales, Hung Le Pham and Paul Ramsden. Published in Dissertationes Math. Volume 498 (2014) 53pp.
- "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.
- "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.
- "Completely positive multipliers of quantum groups". Published in International Journal Mathematics volume 23, no 12 (2012). This paper is available on the ArXiV.
- "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.
- "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.
- "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.
- "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.
- "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.
- "Multipliers, Self-Induced and Dual Banach Algebras", published in Dissertationes Mathematicae volume 470 2010, 62pp. This paper is available on the ArXiV.
- "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.
- "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.
- "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.
- "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.
- "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.
- "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!.
- "Weakly almost periodic functionals on the measure algebra", published in Mathematische Zeitschrift Volume 265 (2010) 285-296. This paper is available on the ArXiV.
- "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.
- "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!.
- "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.
- "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!.
- "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.
- "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.
- "Dual Banach algebras: representations and injectivity", published in Studia Mathematica Volume 178 (number 3) 2007, 231-275. This paper is available on the ArXiV.
- "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).
- "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.
- "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.
- "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

## 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.

- Original advert [LaTeX Source]
- Summary [LaTeX Source]
- Corrections/Clarifications [LaTeX Source].
- Further examples [LaTeX Source]

## 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.

- Some knots: PDF file and LaTeX source.
- The writhe: PDF file and LaTeX source.
- The bracket polynomial: PDF file and LaTeX source.
- The Jones polynomial: PDF file and LaTeX source. Experts will notice that we massaged the definition a little!

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.

- Handout (PDF) which should be used with the fun JAVA applet at: Cut the knot Geoboard
- More spotted paper (PDF).
- Proof of the theorem.

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.

- A non-commutative notion of separate continuity given to the Analysis seminar of TCD and UCD. About the paper: arXiv 1409.7302.
- Fourier Algebras to Quantum Groups. Pure Maths colloquia November 2014.
- I gave four lectures at the Fields Institute as part of the Theme Period on Operator Spaces, Locally Compact Quantum Groups and Amenability. My talks were an introduction to operator algebraic quantum groups. Videos of my talks can be viewed on Fields Video Archive. Here are local copies of the slides: Talk 1, Talk 2, Talk 3, Talk 4. Here is a list of further reading which appears to have disappeared from the Fields website.
- Almost periodic functionals: Talk given at: Banach Algebras 2013. In no way resembles the next talk...
- Almost periodic functionals: Talk given at 2nd QoP meeting: C*-algebras and Banach algebras.
- Groups meet Analysis: The Fourier Algebra: Colloquium style talk given at York, May 2013. Similar to Kent talk, but hopefully even less technical, as now a mathematics (not pure) colloquia.
- The Haagerup approximation property for discrete quantum groups: Talk at ICMS meeting on C*-algebras.
- Positive definite functions on locally compact quantum groups: Talk at Cergy-Pontoise (Slightly unsatisfactory website).
- "Algebraic quantum groups to compact quantum groups": A white-board talk at NAG 5.
- The Fourier Algebra: Colloquium style talk given at Kent, January 2013. I didn't use all the slides!
- Shift invariant preduals: This was a whiteboard talk at YFAG, so here are my hand-written notes. The paper is available on the arxiv.
- Compactifications of quantum groups: Talk given at Queen's University Belfast, October 2012.
- Completely Positive Multipliers: Talk given at the CEMPI Inaugural Conference, Lille, France, September 2012.
- Analysis in Non-Commutative Mathematics: Talk given at the Young Functional Analysts' Workshop 2012, Oxford, March 2012. Really an introduction to Compact Quantum Groups.
- Shift invariant preduals of group algebras: Talk given at the Banach Algebras 2011 conference in Waterloo, Canada, August 2011. A differing take on the talk below!
- Shift-invariant preduals of $\ell^1(\mathbb Z)$: Talk given to the Analysis seminar at UCD, Dublin, February 2011.
- The Fourier algebra and homomorphisms: Talk given to the Analysis and Algebra seminar at Glagow, December 2010.
- Some operator algebra problems from Quantum Information: Talk given to the algebra seminar at Leeds, November 2010.
- Multipliers and Abstract Harmonic Analysis: Talk given at CAHAS 2010 August 2010. Printable version.
- Quantum groups by way of Operator algebras: Talk given At Belfast university, April 2010.
- Multipliers and the Fourier algebras, and non-commutative L^p spaces: Talk given at Birmingham University, March 2010. A third attempt on a theme!
- Multipliers of quantum groups from Hilbert C*-modules: Talk given at Lancaster University, March 2010.
- Multipliers and the Fourier algebras: Talk given at Nottingham University, January 2010. Somewhat similar to the talk beow.
- Multipliers and the Fourier algebras: Talk given at Banach Algebras 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland. A handout version of the slides can be downloaded here.
- Biprojectivity and compact quantum groups: Colloquium talk given to the Pure Mathematics department as Leeds, April 2009. A handout version of the slides can be downloaded here.
- Weakly almost periodic functionals on the measure algebra [PDF]: Talks given at the CIRM, November 2008. This is very similar to the talk below, so the PDF file is just a printable version of my slides (as given to conference participants).
- Weakly almost periodic functionals on the measure algebra [PDF]: Talk given in Newcastle, October 2008.
- Compactifications and the Fourier algebra [PDF]: Talk given in Glasgow, February 2008.
- Quantum compactifications of the Fourier algebra [PDF]: Talk given at the Fields Institute, Toronto, December 2007. To my horror, they have recorded my talk, so the audio is available for download here.
- Compact Quantum Groups: Series of three talks in Leeds, on the whiteboard, November 2007.
- Uniqueness of preduals [PDF]: Talk given at Banach Algebras 07, in Laval, July 2007. This bears some similarly to the two talks below.
- The predual of $l^1(G)$ [PDF]: Talk given in Leeds, May 2007. This is an extension of the BMC below immediately below.
- How to make the bilateral shift weak$^*$-continuous [PDF]: Talk given at the BMC in Swansea, April 2007.
- p-operator spaces and Figa-Talamanca-Herz algebras [PDF]: Talk given in February 2007, in Oxford and at Texas A+M.
- Operator spaces and the Sz-Nagy Similarity Problem [PDF]: Talk given to the junior seminar, November 2006, in Oxford.
- Involutions on algebras of operators [PDF]: Talk given in Glasgow, May 2006.
- Dual Banach algebras [PDF]: Talk given at the British Mathematics Conference, April 2006, in Newcastle.

## 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

- arXiv:1308.1073 [math.FA]: "The approximation property implies that convolvers are pseudo-measures", a write-up of a result of Cowling's, joint with Nico Spronk.
- 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.