Combinatorics in Scotland, group theory in Portugal

I never really wanted to retire. For various reasons which no longer matter, I decided to retire from my position at Queen Mary, University of London, on turning 65 two years ago. I hoped that I would find enough to do to keep me busy, and I have not been disappointed!

Indeed, I have substantially overdone things, and yesterday began teaching my fourth course this year …

Combinatorics in Scotland

This semester I have been teaching a course on Advanced Combinatorics at St Andrews. When they asked me for a syllabus for such a course, I provided three (roughly, one on enumeration, one on finite geometry, and one on group actions), and suggested that one of them could be approved and I would teach that. Instead, they approved all three, so I had to decide which one to teach (and I have the option of teaching the other two in successive years).

So this year it is enumeration. I felt fairly secure, having taught a much briefer course on enumeration at the London Taught Course Centre last semester. But, inevitably, this course is turning out a bit different. Something about the set-up encourages me to ramble on, and tell stories, about Paul Erdős, or Alan Sokal, or someone else. I spent much longer on basics of formal power series, since the class seemed to be well prepared in Analysis, and needed convincing that it really didn’t matter if the series converge or not. (My answer to that is in two parts: the good news is that it doesn’t matter, since the formal power series form a ring in which various other operations, such as substitution, infinite sums and products, and differentiation, all work fine (sometimes under specified conditions), and our manipulations are valid in this setting without the need for convergence; the good news, on the other hand, is that any identity between convergent power series expressed in terms of these operations is also valid in the setting of formal power series, so we can just read off from Analysis all properties of, for example, the exponential and logarithm functions that we need.)

I varied enumeration with a small dose of combinatorics of subsets early on, basically Ramsey’s theorem and Steiner systems, the second giving the opportunity to mention Peter Keevash’s result. (These are two topics where some of the formulae involve binomial coefficients, which had been discussed in the first part of the course.)

In fact, it has not quite gone as I expected. We have talked about subsets but not yet about partitions or permutations; we haven’t done any asymptotics yet, or group actions, or unimodality, and probably several of these topics are not going to get in. Yet I included things like the counting proof of the existence of finite fields, and the inclusion-exclusion formula for the chromatic polynomial of a graph; and if time permits I will do Richard Borcherds’ wonderful proof of Jacobi’s Triple Product Identity by counting states of Dirac electrons.

We are about three-quarters of the way through the course now. It is Spring break, so I have a couple of weeks to get myself ahead and produce the last few instalments of notes.

Group theory in Portugal

Meanwhile, João Araújo persuaded me to teach a course on Group Theory at the Open University in Portugal. The course started yesterday; little has happened yet apart from the students starting to introduce themselves. Already it is clear that I am not very competent with the Moodle interface (well, I did know that already – of course in St Andrews, the first thing I did was to make a course web page!), but I have two competent pairs of hands on the spot to catch me if I fall.

I like the idea of teaching open university students. If one can make invidious comparisons, they are on the whole more committed and enthusiastic than regular university students, since they are making very big sacrifices to study in their own time. I am, of course, a bit nervous that, being quite far away and not speaking their native tongue, I will give them a less good course than they deserve.

But so far, I am fairly happy with the notes I have produced. I started off knowing that there were certain things that should be put in, but after a while I came to see that putting in the things that I would like to be put in would work better. Group theory is a very technical subject; I am trying to give them an overview of the things I am interested in (of course, this means quite a big dose of group actions) without getting too bogged down. So, for example, I will cover doubly transitive groups but may not get on to saying much about primitive groups. (With doubly transitive groups, you get quite quickly to some pretty things.)

Many of these students have a strong background in computers, either a first degree or a job in IT, and several of them are skilful programmers. Maybe I will be able to set them some projects with real content, and get some good stuff done!

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About Peter Cameron

I count all the things that need to be counted.
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