Last week, in the second week of Spring break in St Andrews, I was in Vienna, giving a course of lectures to the PhD students, at the invitation of Tomack Gilmore, a Queen Mary undergraduate now finishing his PhD with Christian Krattenthaler at the University of Vienna.
The lectures were titled “Permutation groups and transformation semigroups”, but didn’t cover everything that can be said about those topics; my aim was to present an exposition of some of the work I have been doing with João Araújo in Lisbon for the last nearly ten years. I lectured twice a day for five days, and so the course naturally fell into five parts. I assumed some knowledge of group theory, but the first day was an introduction to semigroup theory, including both standard material such as regularity and idempotent generation, the analogues of Cayley’s theorem for semigroups and inverse semigroups (the latter is the Vagner–Preston theorem), and the condition for a transformation to have a power which is an idempotent of the same rank, as well as odd extras such as the results of Laradji and Umar constructing inverse semigroups whose orders are central binomial coefficients and Bell and Catalan numbers. The second day was an introduction to the theory of permutation groups, with the basic reductioin theorems as far as the O’Nan–Scott theorem, and a very brief discussion of multiply transitive groups.
The third and fourth days were the heart of the course. Day 3 covered synchronization: the Černý conjecture, the characterisation of non-synchronizing semigroups in terms of graph homomorphisms, the definition and basic properties of synchronizing permutation groups. Day 4 concerned conditions on a permutation group G which guarantee that, for any map s of given rank k, the semigroup generated by G and s, with the elements of G removed, is regular or is idempotent-generated. The condition for the first is the k-universal transversal property of G (that, given any k-set A and k-partition P, there is an element of G mapping A to a transversal for P. This condition is necessary for idempotent-generation of the semigroup (it is necessary and sufficient for the existence of an idempotent with rank equal to that of s), but not sufficient. In general we do not have a combinatorial equivalent of idempotent generation, but in the case k = 2 we do: it is the road closure property, which I have discussed here before.
The final day dealt with miscellaneous topics: automorphisms of transformation semigroups, lengths of chains of subgroups or sub-semigroups, and separating permutation groups. The notes also include a bibliography of books and papers as well as a number of open problems. (The notes are here.)
Doing all this in a week would have been challenging enough, but as well I gave a 90-minute seminar talk on orbital polynomials, a colloquium on the random graph, and a “junior colloquium” on the ADE affair. So quite a busy week, and I am afraid that other jobs had to be put on hold temporarily.
With all this there was not a great deal of time to see Vienna. Though it was early spring, the weather at the weekend was not kind, cold and rainy, though during the week it was better, and the blossom had come out by the time we left.
So much of the sightseeing was indoors. I will mention just the most astonishing thing I saw. Among other galleries, we went to the Academy of Fine Arts (part of the outside is shown above). The picture gallery in the Academy has the famous tryptich on the Last Judgment by Hieronymus Bosch. We think of his work as being mostly either of people doing unspeakable things to each other in gardens, or demons doing unspeakable things to people in Hell; this one certainly has plenty of that, with toads frying sinners in large frying pans or stirring them up in cooking pots. But the real surprise for me was on the back. The tryptich was hung with the side panels folded very slightly forward so that, with care, you could see the painting that would be shown if the panels were closed. This was completely different. The left-hand panel showed St James on a pilgrimage (probably to Santiago de Compostela), taking up most of the panel. He was dressed in dull blue robes and the mountainous landscape behind was in very subdued blue-grey, and the image looked forward to a later period of painting, being an astonishing portrait, of whom I don’t know. The right panel depicted a saint from Ghent giving alms to the poor, in even more subdued style.
The Danube has four branches in Vienna, three of which I saw: the regular river, the old, the new, and the Donaukanal. The last of these is not a canal, nor an open sewer (the meaning of German kanal according to Wikipedia), but a branch which has always existed and was “controlled” in 1598. Unlike the main river, it flows near the centre of town. I don’t know the story of the Alte Donau, which seems to be disconnected now and consist of a series of lakes. The Neue Donau was built for flood control after a serious flood in 1954, though it took a while to reach the decision to build it; work began in 1972 and was finished in 1988. Between the Donau and Neue Donau is a long, straight, and very thin island (which no doubt is crowded in the summer, but when we were there everything was closed and there were only a very few joggers and cyclists to be seen; a sad look to everything).