Peter Cameron's homepage

Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)

I am a quarter-time Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London. In addition, I am an associate researcher at CEMAT, University of Lisbon, Portugal.

I am a Fellow of the Royal Society of Edinburgh.

About me My official page My diary zbMATH profile, MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Biographical stuff, travel diaries On this site Talks, publications, conjectures, coauthors, students Books: Introduction to Algebra, Combinatorics, Notes on Counting, Permutation Groups, Sets, Logic and Categories (some dead links to be sorted out) Problems from my QMUL homepage

Elsewhere My homepage at QMUL Papers on arXiv Algebra and Combinatorics at St Andrews British Combinatorial Committee, BCC Conference List St Andrews pure mathematics seminars CIRCA ORCiD Lecture notes Australasian Journal of Combinatorics Cameron Counts Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

Research snapshots

Bases

A base for a permutation group is a sequence (a₁,…,a_r) of points in the permutation domain whose pointwise stabiliser is the identity. A base is irredundant if no point in the base is fixed by the stabiliser of its predecessors; it is minimal if no point in the base is fixed by the stabiliser of all the others.

A couple of things are clear:

• any minimal base is irredundant (but the converse is false);
• any re-ordering of a minimal base is minimal (but this is not true for irredundant bases).

In 1995, Dima Fon-Der-Flaass and I showed that the three conditions on irredundant bases, namely

• they are preserved by re-ordering;
• they all have the same size;
• they are the bases of a matroid;

Recently several people (Murray Whyte, Scott Harper, Pablo Spiga, maybe others) asked whether the set of sizes of irredundant bases form an interval. This is true. However, the analogue for minimal bases is false. For any integer d > 1, there is a group whose minimal bases have sizes 1 and d only. However, it is not yet known whether they do form an interval if the permutation group is required to be transitive.

A further question is: What happens for greedy bases? One of these is found by choosing the next base point in an orbit of maximum size for the stabiliser of its predecessors. (There is some variability because there might be a number of orbits of the same size.)

Old research snapshots are kept here.

I am Honorary Editor-In-Chief of the Australasian Journal of Combinatorics, an international open-access journal published by the Combinatorial Mathematics Society of Australasia.

A problem

A (total) ordering of a finite vector space is called compatible if every order-preserving bijection between subspaces of the same dimension is linear.

It is straightforward to show that every finite vector space has a compatible ordering: for example, choose any ordering of the finite field F; then the lexicographic ordering of Fⁿ is compatible, for any n.

Problem: How many compatible orderings does the n-dimensional vector space over the field of order q have? (Formulae are known for the cases n = 1 and n = 2.)

Old problems are kept here.