Lukecin is a really small town. According to Wikipedia it has 120 inhabitants but I think almost all of them must live underground because one hardly ever sees someone in the streets. I felt very isolated the whole day unable even to ask the receptionist if I could have dinner in the pension (the town was deserted and I had eaten nothing since the previous night). In the early evening two French men arrived and they could talk to her in German. I nearly hugged them when they told me that I could actually have dinner. The two French men turned out to be the lecturers, Michel Brion and Nicolas Ressayre. Part of my confusion about dinner was due to the fact than in Eastern Europe breakfast and dinner look pretty much the same: some pork salami, tomato and bread with butter, maybe a small appetizer, jam and tea. I was messed-up with my sleep and anyways there was little to do in Lukecin (the only bar closed at 7pm) so I went to sleep at 9pm. I woke up when my roommate arrived. A very funny Argentinian student called Javier. Of all the Spanish-speaking cultures, I find the Argentinian the funnier, with a very measured sarcastic humour. We were laughing till nearly midnight. I also got sick. Apparently tap water in Poland is not drinkable!
The school was quite intense. We had two lectures a day for 3 hours altogether and a 2 hours exercise session. Brion’s lectures were in classical Geometric Invariant Theory. For those who are not expert, we could say that GIT studies quotients of varieties by groups. An algebraic variety X is always a topological space. If a group G acts algebraically on X (i.e. each element is an algebraic automorphism of the variety) in particular it acts homeomorphically and the quotient X/G is another topological space. However in general it is not a variety, and even if it is, hardly ever one obtains a very interesting one (the dimension usually decreases a lot, for instance).
The main problem for it to be a variety has to do with its ring of regular functions R. Since G acts on X, it naturally acts on R. The ring of regular functions of X/G should be R^G, i.e. those elements in R which are invariant by G. This is a subring of R, but surprisingly it is not necessarily finitely generated. In the second ICM, David Hilbert presented a list of ‘fundamental’ problems in Mathematics. Proving that R^G was finitely generated was one of them, the number 14th. However it turned out to be false, as Nagata proved in the 60s. The field of Geometric Invariant Theory (GIT) was therefore initiated by Hilbert and developed by several people. However the two main contributors would be Nagata, and very especially David Mumford, who established criteria to decide when R^G was finitely generated and when Spec R^G coincided with X/G (obviously he also did this in more general settings such as projective and abstract varieties), as well as which subsets of X admitted these quotients. Brion managed to explain these ideas very clearly. I should add what was Mumford’s motivation to develop these techniques. This is the subject of moduli theory, in which one would like to parametrise a set of objects, for instance all curves of a given genus, creating a variety with them. But then those objects which are equivalent (for instance via isomorphism) should be parametrised just by one point. For this it is necessary to have a group acting on the variety and the quotient should be the desired parametrisation.
The subject of Ressayre lectures had to do with applications of GIT to Schubert calculus problems and I found it harder to follow, although he gave a very good explanation of how the Hilbert-Mumford criterion works.
The school had a nice break on Wednesday afternoon, when we were taken to see a Viking village (obviously reconstructed) some amazing views of the Baltic Sea and a conservation park with European deers and bisons. At least 3 of the nights there was a bonfire and the last one we had a barbecue in which we were cooking our sausages in a stick. The last night we also had a walk into the beach (it was just 15 minutes away) and we were very impressed by the amount of stars we could see, something impossible in Britain.
I could not finish the Lukecin experience without mentioning an amusing moment. There is no access to Internet in Lukecin, neither in the pension nor in the cafe. However at some point someone found out that there is a massive hotel complex in the middle of the forest which had unprotected Internet. It was certainly hilarious to see a bunch of people in the middle of nowhere, using their batteries as much as possible in order to send e-mails, check the news and talk to their families.
All in all it has been a great experience. I hope I have another chance to go to a school in there. Jarek Wisniewski, the organiser promised another one for next year. He was wearing a ‘BIG AND NEF’ t-shirt. I keep my fingers crossed 🙂