Reading my blog you could've figured out that we are having a big conference in Antwerp. Yesterday was too hectic to liveblog this, as the neverending author noticed, so I guess I'll just do one post every two days.

Monday September 19

Ragnar Buchweitz quoted the following joke that Matt Satriano (he heard it from Dick Gross) made last week in Banff:

A good talk should contain one joke and one proof, and you should be able to tell them apart.
I think Ragnar turned the punchline into and they should not be the same, which works equally well. Taking this advice from Ragnar (or Matt, or Dick) I might focus my discussion a little bit on the jokes the speakers made, rather than give a detailed outline of what they talked about.

On a more mathematical level, he described Orlov's theorem describing $\mathbf{D}^{\mathrm{b}}(\operatorname{qgr}A)$ as the most important result in algebra since Hilbert's syzygy theorem.

Michael Wemyss's joke was about surgeries on varieties:

Unfortunately, this is algebraic geometry and not topology, so there are rules.

The Scottish accent in which this joke must be delivered is something you have to imagine yourself. Mathematically his talk was really cool, and you can look at his very pretty slides yourself.

Goncalo Tabuada give a really nice talk about some work I talked about earlier on my blog. Using the equivalence of various of the standard conjectures with their noncommutative analogues, it becomes possible to prove them for various high-dimensional varieties using results from low-dimensional varieties as long as they are related via homological projective duality. Cool!

Tuesday September 20

Alexander Polishchuk defined the Hochschild cochain complex and decided to use the notation $\mathrm{CH}$ for it, admitting that it might be confused with the Chow group, only to start referring to it that way afterwards on a few occasions! Besides screwing with the audience's mind in this way, he gave a very nice talk on moduli of $\mathrm{A}_\infty$-structures, exhibiting isomorphisms with other moduli spaces in various examples.

Tobias Dyckerhoff gave an intriguing and seemingly far-reaching categorification result, that was later used by Kapranov in his talk. Tobias talked about Dold–Kan correspondence, and used the letters A and B to denote objects on respective sides of the correspondence, which led Alexander Kuznetsov to point out that he was actually talking about the A- and B-side in mirror symmetry. Tobias remarked that he didn't know how far that thought would actually bring you.

Another cool talk from today was by Henning Krause. I learned that the stable module category and the bounded derived category of coherent sheaves on a (non-constant) finite group scheme does not have Serre duality, but he fixed this by showing it has local Serre duality, i.e. that a certain subquotient associated to a homogeneous prime ideal of the Ext-algebra has Serre duality. I should probably figure out a bit more about non-semisimple cocommutative Hopf algebras to really understand to which kind of examples this applies.

He started his talk with a modification of a quote by Emmy Noether in his ICM address from 1932, which went like

Über die Bedeuting des Kommutative für des Nichtkommutative.

meaning that he wanted to discuss the meaning of commutative ideas to the noncommutative world (Noether said it in the opposite way). Maybe I'll dig a little deeper into the actual context of this quote later.

I'm afraid the first two afternoon talks were a bit over my head, but Ed Segal gave a really nice talk on homological projective duality, or rather Hori duality which is a duality predicted by physicists, and made mathematically rigorous by Jørgen Rennemo and him in a preprint called Hori-mological projective duality. He admitted that the title was his idea. The mathematics in there was of the type I really like: bounded derived categories of smooth projective varieties, but in a setting out of this comfort zone because he needed Artin stacks and subcategories of their derived categories.