This is in a sense a follow-up to my earlier post here, describing how the 6-month programme at the Isaac Newton Institute had come to a premature end because of the covid pandemic.
The time I spent in Cambridge then (early January to mid-March) was a very good time for me. With Rosemary Bailey, Michael Giudici and Gordon Royle, I finished a paper on the subgroup of a transitive permutation group generated by derangements, which was published in the Journal of Algebra. More significantly, I worked with Rosemary, Cheryl Praeger and Csaba Schneider on the geometry of diagonal groups. This was intended as a continuation of the work by Cheryl and Csaba on the geometry of wreath products, published as a book in the LMS Lecture Note series. At the end of our time in Cambridge, we had almost completed a “synthetic” approach to the geometry, and agreed on how to write it up. But in the last couple of days, I had a vision of what the “axiomatic” version should say, although I doubted my ability to prove it. Nevertheless, stuck at home in St Andrews by the lockdown, Rosemary and I were able to prove the result I had in mind. Instead of putting a group in at the start, we were able to make it emerge naturally from purely combinatorial axioms, just as a division ring does in the axiomatic theory of projective planes. The paper has just been published in the Transactions of the American Mathematical Society, and I consider it one of my best.
Now that lockdown restrictions have been lifted, the INI offered the organisers a 3-month period from May to July this year, so I returned to Cambridge for this. However, things did not work so well. Instead of two programmes running, with the possibility of real interactions between the two sets of participants, there were four programmes, two of which were mostly stuck in a building not at all suitable for intensive mathematical research in the back of Churchill College. The organisers found me a share in a large office at the INI, but when I was there I had fewer chances of interacting with other participants. So I divided my time between three places (the third being a flat in Benians Court which had decent wi-fi; it was very tempting just to work there).
I did get a lot of work done during this period. But unlike the first period, it was not with other participants in the programme, but mostly with co-authors on the Graphs on Groups project, mostly in India; also with Marina in St Andrews who was on an undergraduate research internship.
Of course, because of Rosemary’s continuing difficulties after her fall, I had to arrange for her to be in Cambridge with me. Taking her out for walks around the streets, I became very aware of some of the drawbacks of Cambridge. Cyclists there are allowed to ride on the pavement beside the road in many places, while elsewhere they just do it anyway without permission. Often there is little warning to pedestrians about this. (On the stretch along Madingley Road from Storey’s Way to Lady Margaret Road, there was just one sign declaring it a dual-use path, cunningly hidden in the trees and bushes where I was not aware of it for the first month.) Not one in ten Cambridge cyclists seem to have come across a really useful invention, the bicycle bell, which can be used to warn pedestrians of their approach. Many are staring at their phones, heedless of pedestrians. Twice I saw, right outside our flat, a cyclist come off his bike at considerable speed; if this had happened while Rosemary was hobbling along with her stick, the consequences could have been very serious. I was also spooked by the Madingley Road/Lady Margaret Road junction, where there is no part of the traffic light cycle which is safe for pedestrians, and the car which is going to run into you is probably coming from behind. I had to brave this junction many times because the nearest food shop is in the centre of Cambridge, quite a distance from the flat.
The only time the programme came alive was during the last week and a bit. On Friday of the penultimate week there was a memorial day for Jan Saxl, though the location was not well publicised. Then in the last week, we had a closing workshop with some very lovely talks. These included
- Tim Burness on “Base 2 permutation groups and applications” – a strange and brave idea by Jan Saxl, since I would have thought that classifying groups with base size greater than 2 would have been an easier task.
- Scott Harper on “Invariable generation and totally deranged elements of simple groups”. Every finite simple group has a pair g, h of elements which invariably generate it, meaning that you can replace each of them by an arbitrary conjugate and still have a generating set. There is no conjugate pair which invariably generate, since this would imply that a single element generates the group; but Garzoni asked whether there was a pair in the same conjugacy class in Aut(S), for a simple group S, which invariably generate. Scott showed that this does happen, and moreover gave a complete classification of this situation.
- Persi Diaconis on “Double coset random walks on groups” showed us how some classical random walks (starting with contingency tables) can be realised as walks on the double cosets of finite groups, and how the character theory of the group can be used to analyse these.
- Alex Lubotzky on “Good locally testable codes”. The third time I have heard this, but each time it sinks in a bit further. The idea is very simple. Cayley graphs are either left or right: the connection set multiply elements on one side, so that multiplying on the other side gives an automorphism. They combine these two methods by having a left and a right connection set, to give a 2-dimensional Cayley complex with cells (g,lg,gr,lgr) where l comes from the left connection set and r from the right.
The conference honoured Michael Aschbacher and Robert Guralnick, neither of whom was present; both spoke remotely by Zoom. I am getting old and grumpy, and beginning to think that despite all the emissions Zoom talks save, they are not really a substitute for the real thing. But best wishes to both.