In the last week of August, I attended for the first time a virtual conference. This was the 2020 Ural Workshop on Group Theory and Combinatorics, organised by Natalia Maslova at the Ural Federal University in Yekaterinburg and her colleagues. The conference was held as a Zoom meeting, and ran with only one hitch. As fate would have it, it was Natalia’s talk that was disrupted by a technological failure, so she started ten minutes late and had to talk fast. My co-author and St Andrews student Liam Stott was talking in the other parallel session immediately afterwards, so I switched as quickly as I could, only to find that the chair of that session had started him early (I assume the previous speaker hadn’t shown up), and he was three-quarters of the way through his talk already. Fortunately I knew what he was talking about!
Yekaterinburg is four hours ahead of St Andrews, so we had a week of very early rising; we had lunch at 10am, and were finished for the day (in both senses) by 2pm most days.
There were some very enjoyable talks, and as usual I can only mention a few. Cheryl Praeger talked about totally 2-closed finite groups. A permutation group G is 2-closed if every permutation preserving all the G-orbits on ordered pairs belongs to G. Cheryl and her colleagues call an abstract group 2-closed if every faithful transitive permutation representation of it is 2-closed. These groups were first studied by Abdollahi and Arezoomand, who found all nilpotent examples; with Tracey they subsequently found all soluble examples. Now this team augmented by Cheryl has considered insoluble groups. At first they found none, but they found that in fact six of the 26 sporadic simple groups (the first, third and fourth Janko groups, Lyons group, Thompson group, and Monster) are totally 2-closed. Work continues.
We had a couple of plenary talks about axial algebras; Sergey Shpectorov and Alexey Staroletov explained what these things (generalised from the Griess algebra for the Monster) are, and what the current status of their study is.
Greenberg’s Theorem states that any finite or countable group can be realised as the automorphism group of a Riemann surface, compact if and only if the group is finite. Gareth Jones talked about this. The proof, he says, is very complicated. He gave a new and much simpler proof; it did less than Greenberg’s Theorem in that it only works for finitely generated groups, but more in that the Riemann surface constructed is a complex algebraic curve over an algebraic number field.
Misha Volkov gave a beautiful talk about synchronizing automata. He began with the basic stuff around the Černý conjecture, which I have discussed before, but added a couple of things which were new to me: a YouTube video of a finite automaton taking randomly oriented plastic bottles on a conveyor belt in a factory and turning them upright; and the historical fact that the polynomial-time algorithm for testing synchronization was in the PhD thesis of Chinese mathematician Chung Laung Liu (also transliterated as Jiong Lang Liu), two years before the Černý conjecture was announced. Then he turned to new results, and showed that, with only tiny changes (allowing the automaton to have no transition for some state-symbol pair, or restricting the inputs from arbitrary words to words in a regular language) the synchronization problem can jump up from polynomial to PSPACE-complete!
Alexander Perepechko gave a remarkable talk, connecting the Thompson group T, the Farey series, automorphism groups of some affine algebraic surfaces, and Markov triples, solutions in natural numbers to the Diophantine equation x2+y2+z2 = 3xyz. (There is a long-standing conjecture that a natural number occurs at most once as the greatest element in some such triple. The sequence of such numbers begins 1, 2, 5, 13, 29, 34, 89, …. I will not attempt to explain further.)
Rosemary became the fourth author of the “diagonal structures” quartet to talk about that work, which I discussed here. She concentrated on the heart of the proof, the first place in the work where the remarkable appearance of algebraic structure (a group) from combinatorial (a Latin cube with a mild extra hypothesis) appears. Without actually describing how the hard proof goes, she explained the context and ideas clearly. I think this ranks among my best work; and all I did, apart from the induction proof of which Latin cubes form the base, was to insist to my co-authors that a result like this might just be possible, and we should go after it.
One of my early heroes in group theory was Helmut Wielandt; his book on permutation groups was my first reading as a graduate student. Danila Revin gave us a Wielandt-inspired talk. Wielandt had asked, in Tübingen lectures in the winter of 1963-4, about maximal X-subgroups of a group G, where X is a complete class of finite groups (closed under subgroups, quotients and extensions). Let kX(G) be the number of conjugacy classes of maximal X-subgroups of G, Wielandt said that the reduction X-theorem holds for the pair (G,N) if kX(G/N) = kX(G), and holds for a group A if it holds for (G,N) whenever G/N is isomorphic to A. Wielandt asked for all A, and then all pairs (G,N), for which this is true; this is the problem which Danila and his co-authors have now solved.
(I hope Danila will forgive me an anecdote here. At an Oberwolfach meeting in the 1970s, one of the speakers told us a theorem which took more than a page to state. Wielandt remarked that you shouldn’t prove theorems that take more than a page to state. Yet the solution to his own problem took nearly ten pages to state. I think this is inevitable, and simply teaches us that finite group theory is more complicated than we might have expected, and certainly more complicated than Michael Atiyah expected. Indeed, in the very next talk, Chris Parker told us about work he and his colleagues have done on subgroups analogous to minimal parabolic subgroups in arbitrary groups. This is intended as a contribution to revising the Classification of Finite Simple Groups, and they hoped to show that with an appropriate list of properties only minimal parabolics in groups of Lie type and a few other configurations could arise; they obtained the full list and were rather dismayed by its length, which would make the applications they had in mind very difficult.)
Among other fun facts, I learned that the graph consisting of a triangle with a pendant vertex is called the paw in Yekaterinburg, but is the balalaika in Novosibirsk.
On the last day of the seven-day meeting, we had two talks on dual Seidel switching, by Vladislav Kabanov and Elena Konstantinova, who were using it and a more general operation to construct new Deza graphs and integral graphs.
After a problem session, the conference ended by a virtual tour of Yekaterinburg (or Sverdlovsk, as it was in Soviet times), covering the history, architecture and economics, and illustrated by photographs and historical documents; the tour guide was Vladislav’s daughter.
Life was made more difficult and stressful for me because I was doing something which would have been completely impossible in pre-COVID times: I spent some time moonlighting from the Urals conference to attend ALGOS (ALgebras, Graphs and Ordered Sets) in Nancy, France, a meeting to celebrate the 75th birthday of Maurice Pouzet, which I didn’t want to miss. Many friends from a different side of my mathematical interests were there; as well as Maurice himself, Stéphan Thomassé, Nicolas Thiéry, Robert Woodrow, Norbert Sauer, and many others.
The three hours’ time difference between Yekaterinburg and Nancy meant that there was not too much overlap between the two meetings, so although I missed most of the contributed talks in Nancy, I heard most of the plenaries.
Stéphan Thomassé talked about twin-width, a new graph parameter with very nice properties. Given a graph, you can identify vertices which are twins (same neighbours) or nearly twins; in the latter case, there are bad edges joined to only one of the two vertices; the twin-width is the maximum valency of the graph of bad edges. Bounded twin-width implies bounded treewidth (for example) but not conversely; a grid has twin-width 4. Graphs with bounded twin-width form a small class (at most exponentially many of them), and, remarkably, it is conjectured that a converse also holds.
Jarik Nešetřil and Honza Hubička talked about EPPA and big Ramsey degrees respectively; I had heard these nice talks in Prague at the MCW, but it was very nice to hear them again.
Norbert Sauer talked about indivisibility properties for permutation groups of countable degree. I might say something about this later if I can get my head around it, but this may take some time. In particular, Norbert attributed a lemma and an example to me, in such a way that I was not entirely sure what it was that I was supposed to have proved! (My fault, not his – it was the end of a long day!)
Nicolas Thiéry gave a very nice talk on the profile of a countable relational structure (the function counting isomorphism types of n-element induced substructures), something to which Maurice Pouzet (and I) have given much attention, and on which there has recently been a lot of progress. (I discussed some of this progress here, but there has been more progress since.) In particular, structures whose growth is polynomially bounded are now understood, due to Justine Falque’s work, and for primitive permutation groups there is a gap from the all-1 sequence up to growth 2n/p(n), where p is a polynomial, thanks to Pierre Simon and Sam Braunfeld.
Unfortunately the conference was running on BigBlueButton, some conference-enabling software which I had not encountered before but which is apparently popular in France. I am afraid that it was simply not up to the job. The second day of the conference saw some talks and sessions abandoned, because speakers could not connect; I could sometimes not see the slides at all, and the sound quality was terrible. I discovered that one is recommended to use Chrome rather than Firefox, and indeed it did work a little better for me, but not free of problems. On this showing I would not recommend this system to anyone.
In particular, a beautiful talk by Joris van der Hoeven was mostly lost for me. I couldn’t see the slides. Joris’ explanations were perfectly clear, even without the visuals, but sometimes I lost his voice as well. The talk was about the connections between different infinite systems: ordinals, Hardy fields, and surreal numbers. In better circumstances I would have really enjoyed the talk.
I hasten to add that the problems were completely ouside the control of Miguel Couceiro, the organiser, and marred what would have been a beautiful meeting.