It has only been a couple weeks since my last posting on this topic, but there’s quite a bit of new news on the geometric Langlands front.
One of the great goals of the subject has always been to bring together the arithmetic Langlands conjectures of number theory with the geometric Langlands conjectures, which involved curves over function fields or over the complex numbers. Fargues and Scholze for quite a few years now have been working on a project that realizes this vision, relating the arithmetic local Langlands conjecture to geometric Langlands on the Fargues-Fontaine curve. Their joint paper on the subject has just appeared [arXiv version here]. It weighs in at 348 pages and absorbing its ideas should keep many mathematicians busy for quite a while. There’s an extensive introduction outlining the ideas used in the paper, including a long historical section (chapter I.11) explaining the story of how these ideas came about and how the authors overcame various difficulties in trying to realize them as rigorous mathematics.
In other geometric Langlands news, this weekend there’s an ongoing conference in Korea, videos here and here. The main topic of the conference is ongoing work by Ben-Zvi, Sakellaridis and Venkatesh, which brings together automorphic forms, Hamiltonian spaces (i.e classical phase spaces with a G-action), relative Langlands duality, QFT versions of geometric Langlands, and much more. One can find many talks by the three of them about this over the last year or so, but no paper yet (will it be more or less than 348 pages?). There is a fairly detailed write up by Sakellaridis here, from a talk he gave recently at MIT.
In Austin, Ben-Zvi is giving a course which provides background for this work, bringing number theory and quantum theory together, conceptualizing automorphic forms as quantum mechanics on arithmetic locally symmetric spaces. Luckily for all of the rest of us, he and the students seem to have survived nearly freezing to death and are now back at work, with notes from the course via Arun Debray.
For something much easier to follow, there’s a wonderful essay on non-fundamental physics at Nautilus, The Joy of Condensed Matter. No obvious relation to geometric Langlands, but who knows?
Update: Arun Debray reports that there is a second set of notes for the Ben-Zvi course being produced, by Jackson Van Dyke, see here.
Update: David Ben-Zvi in the comments points out that a better place for many to learn about his recent work with Sakellaridis and Venkatesh is his MSRI lectures from last year: see here and here, notes from Jackson Van Dyke here.