In the first week of February, I attended an Oberwolfach workshop on Nonarchimedean geometry and applications. It was a great pleasure to attend a conference in person after such a long period of isolation. Thank you to the organizers for making this week so enjoyable! As usual, here are some scattered recollections from the workshop.

- Due to the hybrid nature of the workshop, the talks on several days didn’t start until 2 pm, presumably to accommodate participants in North America, with the final talk scheduled after dinner. This left a huge swath of unscheduled time, from 9 am until 2 pm, which actually turned out to be kind of great. I don’t know if it was the hunger for in-person interaction after 2 years of isolation, but people really seemed to take full advantage of this free time for vigorous discussion and collaboration. I actually liked this schedule better than the usual schedule.
- Best talks: Johannes Nicaise, Lucas Mann, Piotr Achinger, Yujie Xu, Ben Heuer.
- Worst talk: [redacted]
- Categorical Langlands for GL1 = Langlands for mice.
- Random Question 1 (via PS): Let be an open immersion of an affine scheme into a smooth projective variety. Is the complement an ample divisor on ?
- Random Question 1′ (via DC): Let be an open immersion of affine schemes. Is the nonvanishing locus of a section of an ample line bundle on ?
- One recurring theme throughout the week was the p-adic Simpson correspondence, with excellent talks from Ben Heuer and Matti Würthen. Here something quite amusing happened: in an informal conversation on Tuesday, Ben explained the complicated status of p-adic Simpon to me, and stressed that its most optimistic conjectural form isn’t actually written down, because no one wants to be the one to make a false conjecture. But then Matti Würthen
**did**explicitly conjecture this exact statement in his Friday lecture! For the record, the hope is that for a smooth projective variety , there is an**equivalence of categories**from -representations of towards semistable Higgs bundles on with vanishing Chern classes. As Ben stressed to me, this definitely fails if is replaced with a larger algebraically closed nonarchimedean field. Hmm… - One consequence of COVID measures is that seating for meals was not randomized as usual, but rather was fixed for the entire week. I was assigned to eat with Torsten Wedhorn, Bogdan Zavyalov, and François Loeser. This ended up being a really pleasant group to eat with! I already knew Torsten and Bogdan fairly well, but I’d never spoken with François before, and it turns out he’s a totally charming and delightful person. It was especially wonderful to hear him talk about his astonishing achievements in ultra long distance running. One memorable quote: “The first night without sleep is no problem. The second night is… interesting. And the third night… well… this I cannot recommend.”
- My relationship with Hochschild cohomology has gone from “???” to a vague understanding and a desire to learn more. Thanks to DC for some helpful explanations!
- There was much discussion among the younger participants about what Fargues’s categorical local Langlands conjecture should look like with mod-p coefficients (i.e. in the setting). Of course on the spectral side, one expects to see some category of quasicoherent or ind-coherent sheaves on the special fiber of the Emerton-Gee stack for . On the automorphic side, one should have some category of mod-p sheaves on , and the correct category should fall out of the general formalism developed by Mann in his thesis. One tantalizing fact, sketched out during some of these conversations, is that is definitely -cohomologically smooth (in a
**precise**sense), not of dimension 0 as in the case, but of dimension equal to the dimension of as a -adic Lie group. It is surely no coincidence that this matches the expected dimension of the Emerton-Gee stack for .

Another tantalizing observation: the relationship between the Emerton-Gee stack and Wang-Erickson’s stack of Galois representations is perfectly analogous to the difference between the stacks and appearing in AGKRRV.

On the other hand, it also became clear that most of the analysis in Fargues-Scholze cannot carry over naively to the setting of p-adic coefficients, and that many of the crucial tools developed in their paper simply won’t help here. In particular, the magic charts used by FS, which are -cohomologically smooth for all primes , are definitely NOT -cohomologically smooth. This already fails for . Likewise, their “strict Henselian” property should fail badly. New ideas are very much required!

Could you elaborate more about the analogies between the Emerton-Gee stack and the Wang-Erickson stack and LocSys_G and LocSys_G^restr?

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