Report from Oberwolfach

Recently returned from a workshop on “Arithmetic of Shimura varieties.”

• The organizers did a great job choosing the speakers. For one thing, the overlap between speakers this time and speakers at the previous edition of this event was a singleton set, which I think is a reasonable choice. Moreover, the majority of the speakers were junior people, which is also totally reasonable. It was great to hear what everyone is doing.
• Best talks: Ana Caraiani, Teruhisa Koshikawa, Keerthi Madapusi, Sug Woo Shin, Joao Lourenco
• Best chaotic talk with amazingly strong theorems: Ian Gleason
• SWS has advised a very disproportionate number of Alexanders.
• The usual hike wasn’t possible, due to snow in the mountains. Ah well. Instead we hiked along a path parallel to the road. But there was still cake.
• The food was slightly better than usual: they didn’t serve the notorious bread casserole, and one dinner (the polenta thing) was actually really good.
• “I mean, you know Ben. He’s pretty unflappable. But, yeah… [redacted], uh… flaps him.”
• During the workshop, LM and I hit upon a conceptual explanation for Bernstein-Zelevinsky duality, which works both for group representations and for sheaves on , even when ! More on this later.
• “What was the motivation for this conjecture?” “The motivation was that it is true.”
• Some young people have extremely weird expectations for how the postdoc job market should work.
• The notion of “genericity” in various guises, and its relevance for controlling the cohomology of local and global Shimura varieties, was very much in the air. This came up in Caraiani and Koshikawa’s talks, and also in my (prepared but undelivered, see the first bullet above) talk. My handwritten notes are here, and may be of some interest. Conjectures 3 and 5, in particular, seem quite fun.
• Had some interesting conversations with VL about nearby cycles and related topics. Here’s a concrete question: can the results in this paper be adapted to etale cohomology? There are definite obstructions in positive characteristic related to Artin-Schreier sheaves, but in characteristic zero it should be ok.
• During the workshop, Ishimoto posted a beautiful paper completing Arthur’s results for inner forms of odd special orthogonal groups, at least for generic discrete parameters. I was vaguely sure for several years that this was the (only) missing ingredient in proving compatibility of the Fargues-Scholze LLC and the Arthur(-Ishimoto) LLC for and its unique inner form. After reading this paper, and with some key assists from SWS and WTG, I now see how to prove this compatibility (at least over unramified extensions with ). It shouldn’t even take many pages to write down!
• On a related note, shortly before the workshop, Li-Huerta posted his amazing results comparing Genestier-Lafforgue and Fargues-Scholze in all generality!

As always, Oberwolfach remains one of my favorite places to do mathematics. Thank you to the organizers for putting together a wonderful workshop!

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Higher category theory is not a crime

Last week I attended a wonderful Oberwolfach seminar on “Topological cyclic homology and arithmetic”. Aside from learning lots of actual math, I also learned that homotopy theorists have the best jokes. If you don’t believe me, see here or here.

Arithmetische geometrie

Just attended a week-long meeting at Oberwolfach on arithmetic geometry.

• “So did you do this computation like Gauss, or did you use a computer?” – Gabber to Katz

 

• “Let the indices work it out themselves!” – Janssen

 

• “Shouwu, either you’re going to answer my question, or I’m going to hand you over to Ofer!” – Kisin

 

• Katz (telling a story at the beginning of his talk): “… So anyway, after Spencer returned to Princeton, this is how he described the math department at Stanford [where he had just been a professor for a couple years]: ‘At Stanford, they’re still studying the topology of the unit disk!’ ”
  Conrad (from the audience): “Those days are over.”

 

• “We use what I wrote.” – Janssen reassuring Gabber

 

• “So Peter, why did you turn down the breakthrough prize? [pause] I’m only asking because I’m drunk!”

 

• Anon.: “So Ofer, do you come here much?”
  Gabber: [looks down at table, silently moves his finger across it in stepwise motion for 30 seconds] “Seventeen times.”

 

• Two common referees for technical papers on Shimura varieties: Frobenius and Verschiebung.

 

• Me (after writing down the “new” definition of a diamond): “Is that OK, Peter?”
  Scholze (from the back row): “Looks good!”

 

• Zhang: “So Mochizuki is like the Buddha.  He writes his ideas.  He is satisfied.  If you want to understand them, you visit him, you ask him questions, he gives you a little idea, you go away and study.  You have to be a monk.  Have a monk’s approach.”
  Anon.: “Unfortunately, there aren’t very many good monks.”

 

• A “symplectic lifting whatever shit”. Apparently they’re defined in Kai-Wen Lan’s thesis?

 

• Gabber was NOT happy when he heard about Mochizuki’s Gaussian integral analogy.

 

• While eating the horrible bread casserole thing, which Kedlaya, Lieblich and I had mangled pretty badly while serving ourselves:
  Lieblich:”What is this supposed to BE?”
  Kedlaya: “Some kind of croque madame?”
  Nizioł: “Yes, a croque madame.  But I think you guys croqued it.”