Colva M. Roney-Dougal
Senior Lecturer in Pure Mathematics
Director of the Centre for Interdisciplinary Research in Computational Algebra
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
In 2011, Steve Linton, Max Neunhoffer, Richard Parker and I were 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 structures: see the Case for support if you'd like to know more.
I worked for many years with Derek Holt and John Bray to classify the maximal subgroups of the classical groups over finite fields in dimension up to 12. This project is now finished - the book is now available from CUP! I'm currently working with a PhD student, Anna Schroeder, on dimensions 13 through to 15 or maybe even 16.
Over the past few 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). With Nina Menezes and Martyn Quick I showed recently that the finite simple group with the lowest probability of being generated by two random elements is A_6. I'm currently working with my PhD student, Alexandros Efthymiadis on numerical bounds on the random generation of finite simple groups, as a function of the lowest index of maximal subgroup: Martin Liebeck and Aner Shalev have proved that such a function is linear, so the game is to find the constants.
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 my student Hannah Coutts and I developed 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 did some work with Steve Linton 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).