Conferences

In the past I have written accounts of conferences I have attended over the summer. This year, I am so busy (working more than 8 hours a day when I am supposed to be onn holiday) that a brief summary of three conferences will have to suffice.

About 15 years ago, I directed a programme on Combinatorics and Statistical Mechanics at the Isaac Newton Institute in Cambridge. One of the participants was Tatiana Gateva-Ivanova, who got me interested in the set-theoretic Yang–Baxter equation. We wrote a paper about multipermutation solutions of this equation, including a result about derived length, and promised a follow-up which would include a wreath product construction (a dangerous thing to do: it never got written).

I assume that it was as a result of this that I was invited to a conference in Blankenberge, Belgium, on “Groups, Rings and the Yang–Baxter Equation” last month, organised by Leandro Vendramin of VUB and his team.

Blankenberge is a holiday resort on the coast between the ports of Oostende and Zeebrugge; the cranes, and ships heading for the harbour, of Zeebrugge are clearly visible. The town has a pier; the esplanade is just a long row of high-rise holiday apartments. A tram runs along the coast; I had hoped to take the ferry to Oostende and then the tram, but unfortunately no ferries run to Oostende now. The meeting was in a hotel out of town, with a path leading to the beach, and various sports facilities on-site.

As you might expect from the conference title, the majority of people there were interested in the Yang–Baxter equation, though the invited talks were less focussed on this than the contributed talks, I thought.

In the 15 years since I thought about the Yang–Baxter equation, the subject has moved on. The standard opening gambit now is to regard a solution (satisfying a few extra conditions) as a kind of algebra called a “skew brace”, essentially a set with two group operations on it connected by a twisted distributive law. This subject touches on and unifies several other areas; here are two.

One source of skew braces consists of regular subgroups of the holomorph of a group. The original group and the regular subgroup of its holomorph can form a skew brace. I haven’t had time to delve into this, but I think there must be some extra condition required: any group is a regular subgroup of its holomorph (take its right and left regular representations), while the groups involved in a skew brace are more restricted.

The other main connection is with Hopf Galois theory, about which several people spoke. We were told that Hopf Galois theory describes field extensions which are finite and separable; normality is not required. We were also told that Hopf Galois theory solves problems which ordinary Galois theory cannot solve, which makes no sense to me: any finite separable extension has a normal closure which is a Galois extension.

In the first week of this month, it was Portugal, for a conference on “Theoretical and Computational Algebra” in Pocinho, near Foz Côa in the Douro valley. This place is in two world heritage sites, commemorating palaeolithic art (unlike France and Spain, there were no caves, so the remarkable rock art has been exposed to the weather for thousands of years, but the museum displays help pick out the animal pictures) and the terraces of the Douro valley which have grown grapes for wine for centuries. Because of a strike on Portuguese railways, we had to be taken there from Porto by bus, which followed the tops of ridges with spectacular views; but by the end of the meeting the trains were running, and our return followed the river closely for most of the way, with even more spectacular views of the gorges and terraces.

The meeting was in the Portuguese high-performance rowing centre, well-equipped with swimming pool and gym but a little less well equipped for a conference. The place is built of solid concrete with the result that the acoustics are rather poor except in one auditorium, while the wi-fi doesn’t penetrate to the rooms very well. Nevertheless, we had an excellent conference. I was privileged to give a short course of three lectures on “Permutation groups and transformation semigroups”. There was a lot of emphasis on how proof assistants such as ProverX can help mathematicians (and indeed, during the conference, a paper in which I had a small part, describing a website with back-up computational facilities together with a survey including new results on varieties generated by small semigroups, was accepted by the Journal of Algebra). But a very important function of the conference was to mark the retirement of Gracinda Gomes, who obtained her PhD in St Andrews under the supervision of John Howie and ever since has been a driving force behind the flourishing of the Portuguese school of semigroup theory.

Back to London to pick up clean clothes, then off to Southampton accompanying Rosemary to mODa (“Model-oriented design and analysis, and optimal design”, I think, the OD capitalised since they do double duty). At the moment it is necessary for us to accompany each other to conferences since she is not quite self-sufficient when it comes to carrying food to the table, or travelling with luggage, for example.) As you might expect, this was less my cup of tea than the other two, so I will say little about it; I actually took time out from the conference talks and got on with editing the forthcoming book on “Graphs and Groups, Designs and Dynamics”, a record of the lectures at the G2D2 conference in Yichang just before the pandemic struck. As well as the young people, there were many old hands there, including Henry Wynn, in fine form despite recent illnesses.