This week has been rather a rush. As well as the first week of semester (and my first ever experience of on-line undergraduate lecturing), I have been attending the 13th Iranian International Group Theory Conference in Urmia, Iran. (I had to look it up on the map; it is in the very west of the country, in West Azerbaijan province, near the Turkish border.) The main organiser was Mohsen Ghasemi.
Because Rosemary and I were both lecturing on Wednesday, and Iran is 3 hours 30 minutes ahead of us, I was certainly not able to attend all the sessions, though I did do rather better on Thursday than on Wednesday. Here is a very brief sample of what I heard. I’m afraid that, as usual, it will overlaid with various reminiscences and speculations.
Sasha Ivanov talked about locally projective graphs, starting with GL(4,2) and building up to J4. He had a couple of other configurations which have not been fully understood. One seems to give rise to the alternating group of degree 256, the other is still mysterious. If this sounds as if Sasha is still alert to the possibility of a yet unknown sporadic simple group, that seems to be the case. He mentioned an article in the London Mathematical Society Newsletter for January 2021, on Rubel’s problem. In the middle it mentions the Prouhet–Tarry–Escott problem. This concerns two (k+1)-sets of integers which have the same sum of ith powers for i ≤ k. The largest k for which an example is known is 12, and for the known set the difference between the products is 18.104.22.168.22.214.171.124.23.29.31. He remarked that this number resembles the order of a sporadic simple group.
He thought I would probably agree with him. Sorry, Sasha, it looks to me more like a “round” number, a number with more divisors than any smaller number. I don’t know if it is round in this sense, probably not. But typical sporadic groups skip some primes (the Baby Monster, for example, skips the primes between 31 and 47), while round numbers do not. Also the exponents of the primes in a round number are monotonic non-increasing, while this is not so for orders of sporadic groups (O’Nan, for example, has 5 to the first power but 7 cubed).
Still, it is tempting to speculate on a revisionist history, where this example had been discovered in the late 1960s. John McKay would have connected it with something interesting, John Conway would have developed some moonshine associated with it, and a group might have been coaxed into existence.
This reminds me of something that did happen in the late 1960s. I was walking along a street in London, when a van passed me; on the side was written “The Cameron group”, and a 7-digit number. Unfortunately for me it was an odd number.
Another evocative talk for me was by Eugenio Gianelli. He spoke from his office in the University of Florence. It was a really beautiful talk on character theory (proof of a conjecture of Malle and Navarro). But behind him was a poster displaying the finite simple groups in the format of the Periodic Table. (The same poster was used by the Isaac Newton Institute meeting on group theory that I was attending at the start of last year.) Now I visited Florence in February last year to work with Carlo Casolo and Francesco Matucci on integrals of groups (the topic of my conference talk). It was an adventurous trip in a few regards. North Italy was already locked down but Covid had not yet reached Florence, so I was able to make the trip, though storms did their best to stop me: my flight out was a day late because of storm Ciara, and my trip back might have been delayed by Storm Dennis; the pilot flew anyway but it was the bumpiest landing I have had for a long time.
Anyway, they gave me a share of an office in a different building, but I found it better to work in the office which Carlo shared; I spent quite some time sitting there, waiting for inspiration to strike, and staring at this poster.
We got a huge amount of work done in the week, but a month later, Carlo was dead from a heart attack. Francesco and I did our best to sort out the notes he left us with, but we didn’t succeed in understanding everything. So there are open problems there which might have been closed …
Gareth Jones told us about his beautifully simple diagrammatic proof that the modular group has uncountably many maximal subgroups containing no parabolic elements. This means that the Farey graph, familiar to number theorists, is a Cayley graph admitting an action of the modular group in uncountably many different ways. His results extend to other triangle groups too.
He mentioned that this line of research had been started by Bernhard Neumann, whom he described as his mathematical grandfather (his son Peter was Gareth’s and my doctoral supervisor). Bernard also was a pioneer in the material I talked about. Of course Peter died last month, a few days before his 80th birthday.
Last year has left many sad memories. Let us hope that this year will be better.