Profile

Biography

I did my undergraduate degree in Auckland and postgraduate degrees in Toronto and Auckland. After short positions in Aberdeen, Warwick and Durham, I moved to York in 1999.

Research

Overview

I am interested in algebra, topology and combinatorics. Preferably all three at once.

An apartment in a building of type A₃ (from here)

You can find most of my research articles on the arXiv.

Homological methods in algebra:

‌An ongoing project, joint with Paul Turner. This is mainly the cohomology of sheaves on small categories. These play a similar role in homological algebra to that played by modules in other parts of algebra: they provide a unifying framework into which you can shoehorn a variety of differing areas and problems. A lot of this stuff grew out of work that we did on the Khovanov homology of knots and links. Here are some recent lecture notes on Khovanov homology for beginners.

Partial symmetry:

Another ongoing project, joint with John Fountain and more recently James East. If by “mirror symmetry” one means the theory of reflection groups, then by “partial mirror symmetry” one means the theory of reflection monoids. Reflection groups lie at the intersection of many parts of algebra and mathematical physics. Reflection monoids were invented by John Fountain and myself in 2010.

Representation theory:

Quite a few years ago now I proved that every Fuchsian group has among its homomorphic images all but finitely many of the alternating groups. This answered an old conjecture of Graham Higman that had its origins in the study of groups of conformal automorphisms of Riemann surfaces. Here is a very old talk on this and its connection with the congruence subgroup problem. So I have always been interested in the representation theory of infinite discrete groups. More recently, and as an off-shoot of the partial symmetry programme, I have become interested in the representation theory of inverse monoids.  Here are some recent lecture notes on the representations of inverse monoids, written mainly for group theorists.

Geometric manifolds:‌

Steve Tschantz's picture of the 1-skeleton of the 6-dimensional Gosset polytope 2₂₁ (from here)

I work less on this these days, although there a few open problems that still bug me (particularly this old chestnut: what is the minimum volume of a compact hyperbolic 4-manifold?). Over the years I worked with John Ratcliffe, Steve Tschantz and Colin Maclachlan (Aberdeen). Most of this work has been on the volume spectra of orientable hyperbolic manifolds in a given dimension, usually an even dimension, where the presence of an Euler characteristic makes life a lot easier. A highlight was a solution to the Siegel problem in 6-dimensions, with the construction of a minimal volume orientable (but non-compact) hyperbolic 6-manifold. The construction is essentially algebro-combinatorial and uses Coxeter groups.

Research group(s)

Algebra

Available PhD research projects

I am happy to discuss potential PhD projects in any of the above areas.

Supervision

Current PhD students:

• Majed Albaity
• Sanaa Bajri
• Sam Ford

Former PhD students:

• Sajida Koussar
• Jim Woodward 
• Guinevere Dyker

Teaching

Postgraduate

‌

The African Institute of Mathematical Sciences (AIMS) was originally founded in Cape Town, South Africa in 2003 and has subsequently spread to be a pan-African center with Institutes in Senegal, Ghana, Cameroon and Tanzania. The emphasis is on post-graduate training and research for talented students recruited from all over Africa. AIMS brings lecturers from around the world to teach intensive courses. I have been involved with AIMS Ghana since it opened in 2012, visiting five times to teach.

External activities

Invited talks and conferences

(just the ones from this millenium...)

• Higher limits of sheaves, April 2017, BMC Durham.
• Khovanov homology 101, August 2016, AIMS Ghana (I wasn't able to give these in the end...but here they are anyway).
• Higher Limits, February 2016, Southampton.
• Higher Limits, January 2016, Heilbronn Insistute, Bristol.
• The homotopy theory of Khovanov homology, March 2014, Sheffield.
• The homotopy theory of Khovanov homology, February 2014, Glasgow.
• Partial mirror symmetry, December 2013, UEA.
• A Problem of Siegel, June 2012, Fribourg, Switzerland.
• A Problem of Siegel, April 2012, Adelaide, Australia.
• Four Lectures on Buildings, June 2011, Fribourg, Switzerland.
• Knots, posets and sheaves, May 2011, Adelaide, Australia.
• Knots, posets and sheaves, March 2011, Bar-Ilan, Israel.
• Knots, posets and sheaves, December 2010, ICMS, Edinburgh.
• Knots, posets and sheaves, October 2010, Leeds.
• Symmetry and partial symmetry, October 2009, Ilorin, Nigeria.
• Coloured poset homology, May 2009, Leicester.
• Coloured poset homology, March 2009, Durham.
• Coloured poset homology, November 2008, Manchester.
• Partial symmetry, reflection monoids and Coxeter groups, June 2008, Heriot-Watt.
• Knot invariants: natural and not, December 2007, Newcastle colloquium
• Knot invariants: natural and not, November 2007, York.
• Weyl groups, lattices and geometric manifolds, July 2007, Hyperbolic Volume Fribourg, Switzerland
• Partial mirror symmetry: reflection monoids, June 2007, Maxwell Institute, Edinburgh
• Partial mirror symmetry, November 2006, Sydney.
• Coxeter groups and geometric manifolds, October 2006, Institute of Geometry and its Applications, Adelaide.
• Siegel's Problem, March 2005, Cambridge.
• Siegel's Problem, February 2005, Warwick.   
• Coxeter groups and geometric manifolds, October 2004, Sydney.
• Coxeter groups and geometric manifolds, August 2004, Discrete Groups and Hyperbolic Manifolds, Aberdeen. (in honour of the retirement of Colin Maclachlan)
• Coxeter groups and geometric manifolds, July 2004, London Mathematical Society Invited Lecture Series, Southampton.
• Coxeter groups and geometric manifolds, May 2004, Fribourg (Switzerland).
• Coxeter groups and geometric manifolds, March 2004, Southampton.
• Coxeter groups and geometric manifolds, March 2004, British Postgraduate group theory conference, Warwick.
• Coxeter groups and geometric manifolds, February 2004, North British geometric group theory, ICMS, Edinburgh.
• Coxeter groups and geometric manifolds, October 2003, Vanderbilt, Tennessee.
• Coxeter groups and geometric manifolds, October 2003, UT Austin, Texas.
• The Geometry and Topology of Groups, May 2003, Universida Autonama, Madrid.
• A conjecture of Serre, April 2003, Leeds.
• A conjecture of Serre, March 2003: Imperial College (London algebra colloquium).
• Rigidity and Arithmeticity, January-March 2003, York.
• Images of geometric groups, November 2002, Newcastle.
• Images of geometric groups, November 2002, Aberdeen.
• A conjecture of Serre, September 2002, Conference marking the Anniversary of the birth of V I Borevich, St Petersburg.
• Images of geometric groups, July 2001, Aspects of finitely generated groups, Trento.
• Algebraic Geometry over Groups, March 2001, Warwick.