Peter Cameron's homepage

Welcome to my St Andrews homepage. This page is under construction (and probably always will be!)

I am a half-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 MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Short bio, some interviews On this site Teaching page: MT5861, Advanced Combinatorics (next presentation 2020-21 Semester 2) 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)

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

Research snapshots

Diagonal groups (update)

Diagonal groups D(G,m) for G a finite simple group are one of the classes arising in the O'Nan--Scott Theorem. However, they can be defined for any group G, not necessarily finite or simple. With Rosemary Bailey, Cheryl Praeger and Csaba Schneider, I have defined for each pair G,m where G is a group and m a dimension greater than 1, a geometric structure called a diagonal semilattice whose automorphism group is the corresponding diagonal group. We also gave simple axioms for these geometries. For m = 2, a structure satisfying the axioms is precisely a Latin square, but for m > 2 the group G arises naturally from the combinatorial axioms.

These structures consist of m+1 partitions of a set, any m of which are the least elements in a Cartesian lattice. With Rosemary, Cheryl, and Michael Kinyon, I have looked at larger sets of partitions satisfying these conditions, and found connections with orthogonal arrays, arcs in projective space, and MDS codes.

Finally, the paper with Sean Eberhard contains a mistake which we have been unable to fix.

• R. A. Bailey, P. J. Cameron, C. E. Praeger and C. Schneider, The geometry of diagonal groups, arXiv 2007.10726
• R. A. Bailey, Peter J. Cameron, Michael Kinyon and Cheryl E. Praeger, Diagonal groups and arcs over groups, arXiv 2010.16338

Power graphs (update)

There are various classes of graphs which are recognised by a small list of forbidden induced subgraphs, and which have nice algorithmic properties and various applications; these include perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs. With Pallabi Manna and Ranjit Mehatari, I have investigated when the power graph of a group belongs to one of these classes. Power graphs are always perfect; we have found all groups for which they are either split or threshold, and all nilpotent groups for which they are either cographs or chordal.

• P. J. Cameron, P. Manna and R. Mehatari, Forbidden subgraphs of power graphs, arXiv 2010.05198

Automorphisms of shift spaces

In two papers with Collin Bleak and Shayo Olukoya, I have looked further at the connection between strongly synchronizing automata and transducers, the Higman–Thompson finitely presented infinite simple groups, and the automorphism group of the one- or two-sided shift.

• C. Bleak, P. J. Cameron and F. Olukoya, Automorphisms of shift spaces and the Higman–Thomspon groups: the one-sided case, arXiv 2004.08478
• C. Bleak, P. J. Cameron and F. Olukoya, Automorphisms of shift spaces and the Higman–Thomspon groups: the two-sided case, arXiv 2006.01466

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 group G is said to satisfy the finiteness condition F_n if it acts geometrically (that is, properly and cocompactly) on an (n−1)-connected CW-complex. It satisfies F_∞ if it satisfies F_n for all n. Here, F₁ is equivalent to being finitely generated, and F₂ to being finitely presented. Finitely generated free groups satisfy F_∞.

The Houghton group H_n satisfies F_n but not F_n+1. Also, this group contains the finitary symmetric group, so it is highly transitive.

Problem: Given a natural number n, does there exist a group G such that

• G is oligomorphic but not highly transitive (and/or contains no finitary permutation);
• G satisfies F_n but not F_n+1?

Old problems are kept here.