Yesterday was my tenth consecutive day of conferences.
I’ll start this post with a detour. Bill Tutte was one of the deepest thinkers on combinatorics in the twentieth century. One of my favourite mathematics books is his Graph theory as I have known it. What appears on the surface to be a number of chapters on unrelated topics in graph theory actually charts the progress of his mathematical career, with clear explanations of the links and thought processes that brought him from one topic to the next.
His first mathematical work, done (with Brookes, Smith and Stone while they were undergraduates at Cambridge, where Tutte was actually studying chemistry) was on squaring the square, that is, dissecting a square into squares of unequal sizes. Graham Farr showed us the origin of this problem with H. E. Dudeney in the early twentieth century, and a picture of a beautiful realisation of the solution that Tutte and his coauthors found on the Tutte memorial in his hometown of Newmarket near Cambridge.
The National Museums website, linked above, suggests that the squared square represents “Bill’s early fascination with mathematical puzzles”. But, as the graph theory book makes clear, and as Graham stressed in his talk, this was much more than a puzzle: it led Tutte, via electrical networks and Kirchhoff’s laws, to deletion and contraction and so to the Tutte polynomial and some deep insights in graph theory and matroid theory. It was indeed the start of his career, and arguably the event that persuaded him to become a mathematician rather than a chemist.
I was reminded of this yesterday at the final day of a Cambridge conference, celebrating the birthdays of two of my mathematical brothers, Martin Liebeck and Jan Saxl.
I caught just four talks, by Laci Pyber, Aner Shalev, Cheryl Praeger and Peter Neumann.
While the first three told us lots of big impressive theorems (including, among many others, the disproof of a conjecture of Étienne Ghys, the solution of Waring’s problem in finite simple groups, and the proof of a conjecture of Charles Sims), Peter Neumann adopted a different strategy, which he only explained at the end of his talk, and which I am going to discuss here. He told us about a property of finite groups called the TPP (triple product property), devised by Henry Cohn and Christopher Umans in a group-theoretic approach to find new bounds for the exponent of matrix multiplication (the smallest number c such that two n×n matrices can be multiplied with at most nc+o(1) arithmetic operations.
Had they succeeded, this would have been highly significant; but they didn’t, and nobody else has since, so the problem has to stand on its own merit, and probably most people would agree that it is not the next big thing in finite group theory.
However, Peter (speaking as the thesis supervisor of both of the birthday boys, as well as of Cheryl Praeger and me), ended by talking about what is maybe the critical problem in supervising a student: finding a good problem. This should be a problem to which the supervisor does not know the solution, but believes that a reasonably strong student would be able to find it. (I completely agree: the default assumption about a new student is that (s)he will go beyond what I have been able to achieve.) It is also important that the problem will open up pathways leading in various different directions, which a good student will be able to follow. He seemed to regard the TPP problem as one filling this specification, as were the problems he suggested to Martin and Jan when they were students.
So let me conclude with my experience. As an undergraduate in Brisbane, I did a project on the simplicity of the groups PSL(n,q), taken from Dickson’s book (and quite hard going it was with its out-of-date terminology). But I also worked on several problems I devised myself, mostly general mathematics, but often with some combinatorial flavour (although, at the time, combinatorics was not a recognised subject at the University of Queensland). The best worked out was to develop an analogue of cartesian product for unordered n-tuples, that is, put structure on the set of unordered n-tuples of elements of some structured set.
In Oxford, Peter gave me the problem of finding all the primitive permutation groups of degree 57. (A long paper he had just written, on primitive groups of degree 3p for p prime, was to be my guide.) Now it just so happened that the only such group which is not 2-transitive is isomorphic to PSL(2,19); the point stabiliser is isomorphic to PSL(2,5), and acts as such on one of its orbits (having size 6). This 2-transitive action of the stabiliser meant that, in modern terminology, the orbital graph is 2-arc transitive. But at the time, Donald Higman and Charles Sims had only just introduced combinatorial and graph-theoretic methods into the study of finite permutation groups. Another primitive group in which the stabiliser acts 2-transitively on one of its orbits was the recently-discovered Higman–Sims group, which played a big part in my thesis. I described at my 60th birthday conference in Ambleside how I got from the Higman–Sims group to the Urysohn metric space.
So I think that my supervisor’s principle was just right for me.