## Wednesday September 21

Bertrand Toën talked about his approach to a proof of Bloch's conjecture, mentioned before on my blog. Back then it was exciting stuff, and after his overview talk it still is very cool to see how it is possible to apply noncommutative algebraic geometry to problems of an arithmetic nature.

He mentioned that a strong version of resolution of singularities would also settle the conjecture, because of a special case of the conjecture proven by Kato–Saito that would imply the whole thing provided this mythical result. Using noncommutative algebro-geometric techniques seems like a more fun thing to do from my perspective.

I already saw Amnon Yekutieli's talk on noncommutative MGM equivalence last week in Banff, and back then I thought I understood things quite well, but hearing them a second time definitely helps (and Henning Krause's talk the day before was probably a good way of getting in the right setting). You can see his very nice slides on his webpage. I should probably learn more local cohomology.

Continuing (somewhat) the theme of Amnon's talk Leonid Positselski gave a wonderfully set up lecture. Ragnar's advice was to have a proof and a joke, but he rather opted for two proofs of a baby case theorem, which then split into two completely different theorems: Grothendieck duality and MGM equivalence.

## Thursday September 22

After hearing several talks on homological projective duality at various conferences (and finally understanding it somewhat) the lecture by Alexander Kuznetsov was really interesting. It concerned constructing new (and more complicated) homological projective dual varieties out of old ones, using a construction called categorical joins. So far it is still work in progress, but I am very much looking forward to his paper with Alex Perry on this.

The talk by Paolo Stellari was first of all a very good overview of the properties of what he called the Kuznetsov component: the non-trivial part in the semiorthogonal decomposition of a cubic fourfold, whose rationality has featured on this blog several times. I finally have a clear picture of how Kuznetsov's conjecture regarding the non-rationality of the very general cubic fourfold relates to the divisors $\mathcal{C}_D$ on the moduli space and the appearance of (twisted) K3 categories, maybe I'll write it up at some point if it is not already in some introduction of a paper. The cool thing is that so far this correspondence only holds generically, and one of the applications (in progress) of the (proven) existence of Bridgeland stability conditions on the derived category of a cubic fourfold would be to remove these conditions.

The talks by Toby Stafford and Sue Sierra were about their joint work on classifying noncommutative rational surfaces. From Toby I learned how their construction behaves regarding (fat) point modules in actual examples, and I have gained a better intuition for Artin's conjecture, to which I should probably dedicate a blog post at some point. Sue told us her version of the blowing up points on a plane in an airport story: Toby and her were on the same flight from Banff as me, and luckily no-one overheard them discussing blowups during the flight!

The last talk of today was by Shinnosuke Okawa, who is constructing moduli spaces of marked noncommutative del Pezzo surfaces. These objects are something I like to think about too (more on this hopefully soon when the papers are finished), and this was the first time I heard him talk about the results for del Pezzo surfaces of low degree (so with many points blown up). The role of the marking in the noncommutative setting is something I definitely want to understand better, and I'm looking forward to their preprint.