Trying to keep track of everything happening in the Langlands program area of mathematics is somewhat of a losing battle, as new ideas and results keep appearing faster than anyone could be expected to follow. Here are various items:
- Dennis Gaitsgory was here at Columbia yesterday (at Yale the day before). I don’t think either lecture was recorded. Attending his lecture here was quite helpful for me in getting an overview of the results recently proved by him and collaborators and announced as a general proof of the unramified geometric Langlands conjecture. For details, see the papers here, which add up in length to nearly 1000 pages.
For a popular discussion, see this article at Quanta.
To put things in a wider context, one might want to take a look at the “What is not done in this paper?” section of the last paper of the five giving the proof. It gives a list of what is still not understood:
Geometric Langlands with Iwahori ramification.
Quantum geometric Langlands.
Local geometric Langlands with wild ramification.
Global geometric Langlands with wild ramification.
Restricted geometric Langlands for ℓ-adic sheaves (for curves in positive characteristic).
Geometric Langlands for Fargues-Fontaine curves.Only the last of these touches on the original number field case of Langlands, which is a much larger subject than geometric Langlands.
- Highly recommended for a general audience are the Curt Jaimungal – Edward Frenkel videos about the Langlands story. The first is here, the second has just appeared here, and there’s a third part in the works. One scary thing about all this is that Frenkel and collaborators are working on an elaboration of geometric Langlands in another direction (“analytic geometric Langlands”), which is yet again something different than what’s in the thousand-page paper.
- Here at Columbia, Avi Zeff is working his way through the Scholze proposal for a version of real local Langlands as geometric Langlands on the twistor P1, using newly developed techniques involving analytic stacks developed by Clausen and Scholze. This is an archimedean version of the Fargues-Scholze work on local Langlands at non-archimedean primes which uses ideas of geometric Langlands, but on the Fargues-Fontaine curve. Together these provide a geometric Langlands version of the local number field Langlands program, with no corresponding geometric global picture yet known.
- Keeping up with all of this looks daunting. To make things worse, Scholze just keeps coming up with new ideas that cover wider and wider ground. This semester in Bonn, he’s running a seminar on Berkovich Motives, and Motivic Geometrization of Local Langlands, promising two new papers (“Berkovich motives” and “Geometrization of local Langlands, motivically”), in preparation.
As a sideline, he’s been working on the “Habiro ring” of a number field, finding there power series that came up in the study of complex Chern-Simons theory and the volume conjecture. According to Scholze:
My hope was always that this q-deformation of de Rham cohomology should form a bridge between the period rings of p-adic Hodge theory and the period rings of complex Hodge theory. The power series of Garoufalidis–Zagier do have miraculous properties both p-adically and over the complex numbers, seemingly related to the expected geometry in both cases (the Fargues–Fontaine curve, resp. the twistor-P1), and one goal in this course is to understand better what’s going on.
- Finally, if you want to keep up with the latest, Ahkil Mathew has a Youtube channel of videos of talks run out of Chicago.
Actually, geometric Langlands for Fargues-Fontaine curves is being dealt with – see my lectures on Akhil’s channel for a detailed discussion of the current status of things.
“Keeping up with all of this looks daunting.”
And gosh, you’ve been writing about the Langlands program for two decades! At this time is there any component of the Langlands shockwave that pulls focus on physics? (I assume Langlands/QFT is a topic still dear to you)
Quasiparticular,
First of all, several people wrote in, outraged about how mathematical physics has become. This post was purely about developments in mathematics research by mathematicians, not about physics. Physicists are not involved at all in this.
I’ve always been intrigued by tantalizing connections of this field to physics, but these recent developments don’t seem to go in that direction. From its beginnings, there have been relations between geometric Langlands on a Riemann surface and 2d conformal QFT. Witten found a great deal of material relating geometric Langlands to a specific 4d QFT. David Ben-Zvi has written a lot about how the topological QFT point of view illuminates the larger structure of this subject. That the twistor P1 shows up here as the description of an archimedean point and in physics as the twistor description of a space-time point continues to fascinate me (but, may be meaningless..)
So, nothing really new that I’m aware of on the relation to physics front, but, as pure mathematics, a great deal is happening here.
(Math) people might be interested in a conference in Copenhagen last summer, consisting of lecture series by Clausen, Efimov and Nikolaus. “Continuous K-theory, dualizable and rigid categories.” Videos as well as good notes: https://www.math.ku.dk/english/calendar/events/masterclass-continuous-k-theory/ . I’ve only dipped into to the Clausen talks so far, joint work with Scholze. More analytic structure built into complex manifolds.
The second Curt Jaimungal – Edward Frenkel video Peter mentions contains a nice summary of three different Langlands’es, starting with the only one Robert Langlands endorses for the use of his name (the number theory one)! It goes from about the 1:40 mark to the end of the video. I don’t recall it laid out so concisely in his book Love & Math. Somewhere else in the video Frenkel recounts his conversations with Langlands, which is charming. (And informative, are far as leading to making abstract things more like functional analysis, as far as I understand it.)
Two hundred year old number theory problems are a good validation of math, in a sense applied math. Since group theory started there, it makes some sense the algebra would grow to almost incomprehensible levels. As long as it solves problems.