Colva M. Roney-Dougal
Senior Lecturer in Pure Mathematics
Research InterestsMy research is mostly in finite group theory, with a few excursions into theoretical computer science. More recently I've been dipping my toe in the infinite as well. Here's a few more specific details.
Generalisations of small cancellation
Steve Linton, Max Neunhoffer, Richard Parker and I have recently (May 2011) been awarded an EPSRC grant of £442,115.68 for a project called "Solving word problems via generalisations of small cancellation". This project seeks to use the idea of curvature redistribution to produce word-problem solvers for a wide variety of algebraic structres: see the Case for support if you'd like to know more. We will be advertising shortly for a three year postdoctoral position to work on the project, please contact me if you'd like more information.
I'm working with Derek Holt and John Bray to classify the maximal subgroups of the classical groups over finite fields in dimension up to 12. We've almost finished writing our book on this, with any luck it will be complete in the next few months!.
Over the past two or three years, I've become interested both in random generation of permutation groups, matrix groups, and simple groups, and in algorithms for producing random elements of groups (for example, of the derived subgroup of a group given by generators).
For the past many years I've been working off and on with the matrix group recognition problem. I am especially interested in finding new ways of using the algorithms which have been developed in the context of the matrix group recognition project for other purposes. For instance, a few years ago I developed a new algorithm for determining the conjugacy of subgroups of the general linear group, and am currently working with my student Hannah Coutts to develop an algorithm to compute normalisers of subgroups of classical groups, in their natural linear representations. I've also done some work on recognising groups as lying in certain Aschbacher classes, and algorithms for the classical groups such as computing the spinor norm of elements of orthogonal groups.
I've written a whole sequence of papers classifying primitive groups of small degree, most recently up to degree 4095 with Hannah Coutts and Martyn Quick.
This is in collaboration with various people in computer science, mostly at St Andrews. The overall setup is that one has a finite set of variables, each with a finite domain, and there's a collection of restrictions on the values that the variables can hold (graph colouring is a good example here). The general technique is to search for one or all solutions; where I come in is that often the problem can have some symmetry and we then want to reduce search by using the symmetry.
Possibly surprisingly, this grew out the of the work on constraint satisfaction. I'm working with Steve Linton and our student, Yohei Negi, on algorithms for inverse semigroups, and also developing a represention theory for inverse semigroups represented by means of partial bijections (between subsets of some fixed set).