Research

Broadly speaking I am interested in the structure and representation theory of finite groups, and in combinatorial objects upon which such groups may act. Within these areas I am interested in developing computational algorithms and techniques for computing with concrete examples, especially when these examples seem out of reach of the existing methods and technology.

I’ve also got a bit of a side-line in static program analysis. Just for fun.

Find me on MathSciNet (if you have access to it).

Here are some things I am working on:

• Conjugacy Class Representatives in the Monster Group (With R. A. Wilson.) LMS J. Comput. Math. 8 (2005). (MR2166573)
  We use the computer constructions of the Monster to determine words in the generators for an element in each conjugacy class, up to algebraic conjugacy. We show how to distinguish the conjugacy classes. You can(not) access the Monster elements database on-line.
• The Character Table Of A Maximal Subgroup Of The Monster. (With R. A. Wilson.) LMS J. Comput. Math. 10 (2007) 161--175.
  to be precise.
• The Character Table Of A Group Of Shape . (Preprint and program, tar.gz, 124k.) LMS J. Comput. Math. 13 (2010) 82–89
  We use the technique of Fischer matrices to write a program to produce the character table of a group of shape from the character tables of G, G:2, 2.G and 2.G:2. We also supply a simplified proof of a result frequently used in the Fischer matrices method.
• Nets of the Monster. I am working on a classification of “nets”, combinatorial objects obtained from the Monster group. At net is an orbit (under a certain action) of the 3 string braid group on a triple (a,b,c) of 2A-involutions of the Monster. The problem is to obtain representative triples for all such orbits, up to conjugacy in the Monster. An easy structure constant calculation shows there to be about 1·4 million classes of triples. My strategy has been to work in p-local maximal subgroups to obtain all nets centralised by an element of order p. I have completed all cases for p ≥ 5.
• A Trajectory-based Strict Semantics for Program Slicing. (With David Binkley, Sebastian Danicic, Mark Harman, Robert M. Hierons, Àkos Kiss, Mike Laurence, and Lahcen Ouarbya.) Theoretical Computer Science 411 (2010) 1372–1386.
  We define a simple, intuitive trajectory-based equivalence relation which accurately captures the relationship between programs and the slices produced by the most commonly used and well-known slicing algorithms, Weiser’s and the PDG algorithm, even for non-terminating programs. Our new equivalence, unlike other approaches, is based on standard semantics. It deals with non-termination by stipulating that a program and its slice must ‘agree’ in all terminating ‘approximations’ based on finite loop unfoldings. We prove that this equivalence is preserved by Weiser’s algorithm and all slicing algorithms based on data and control dependence. We also demonstrate that this new equivalence more accurately captures the behaviour of slicing algorithms than previous work.
• Generalising Control Dependence. (With Sebastian Danicic, Mark Harman, Àkos Kiss, and Mike Laurence.) (Accepted).
  Weak and strong projection, properties which capture the underlying semantics of control dependence, are defined. Weak and strong commitment-closedness, generalisations of non{termination sensitive and insensitive control dependence to arbitrary finite directed graphs, are introduced and shown to satisfy these desirable semantic properties. Low polynomial{time algorithms for computing these generalised forms of control dependence are given.
  Our formulation is attractively simple and, because of its generality, widely applicable to both existing and future notions of control dependence. To demonstrate this, it is proved that all published forms of control dependence can be implemented using weak or strong commitment-closedness, thereby satisfying either the weak or strong semantics. A by-product of this research has, thus, been to classify all previous forms of control dependence into just two: weak and strong.

I enjoy seminars in the school, including the London Algebra Colloquium, the Monday pure mathematics seminar, sometimes the dynamical systems seminar, and the representation theory study group where I regularly contribute.

I also enjoy conferences! Maybe you’ve seen me at one.

Octad of the week.

A road sign.

Some stuff

Using TrueType fonts in LaTeX.

Using Times or Palatino in LaTeX, including equations.

Placing the end of proof symbol nicely in LaTeX.

Maths notes

Paper models of polyhedra.

Controlling TYPE/CREATOR codes and dealing with evil wicked file extensions in MacOS X.

Autographed Dynkin diagrams.

Compiling the MeatAxe on MacOS X.

Sometimes I make websites for people.

• Johnny Ikon, a big php/mySQL project.
• Farnley Tyas Clay Pigeon Shoot. Sometimes I go, but I don’t shoot.
• GrassRoots Food Network. Quite a big project.
• Taylormade insulated window covers (both internal and external) for camper vans and RVs.

More stuff.

I have finally been ejected into the private sector where I am Head of Spend Analysis. I made a carbon footprint analysis thingamajig too.

Please buy one, or form a limited company!