The Canadian publication *The Walrus* today has a wonderful article about Robert Langlands, focusing on his attitude towards the geometric Langlands program and its talented proponent Edward Frenkel. I watched Frenkel’s talk at the ongoing Minnesota conference via streaming video (hopefully the video will be posted soon), and it was an amazing performance on multiple levels. A large part of it was a beautiful explanation of the history and basic conception of what has come to be known as geometric Langlands. He then went on to explain carefully some of the ideas in the recent Russian paper by Langlands, basically saying that they worked in the Abelian case, but could not work in the non-Abelian case. He ended by describing some alternate ideas that he is working on with David Kazhdan. Langlands was in the audience and at the end of the talk rose to comment extensively, but I couldn’t hear his side of this since he had no microphone (that Frenkel was sticking to his guns though was clear).

Besides giving the talk, Frenkel has made available a manuscript which gives a much more detailed version of the talk. See section 3.5 for an explanation of what he sees as the fundamental problem with what Langlands is trying to do: even in the simpler case of G/B over the complex field, you can’t successfully define a Hecke algebra in the way that Langlands wants.

The conference is finishing up right now, with final remarks by Langlands coming up later this afternoon.

A few more items, mostly involving my Columbia math department colleagues:

- If you connect quickly to the streaming video from Minnesota, you may be able to catch Michael Harris’s talk on local Langlands.
- Quanta magazine has an article about a recent proof of an old conjecture by Dorian Goldfeld about ranks of elliptic curves. This is due to Alexander Smith, now a
~~third~~fourth year graduate student at Harvard (he started working on this while an undergrad at Princeton, with Shouwu Zhang). His twin brother Geoffrey is also a math grad student at Harvard. - Andrei Okounkov has been giving some talks recently at various places about developments in geometric representation theory with some connection to physics, under the title
*New worlds for Lie Theory*. The slides from the ICM version of the talk are here. - For those more interested in physics than mathematics the new issue of Inference has some articles you might enjoy. In particular, Sheldon Glashow is no fan (neither is Chris Fuchs) of the book I reviewed here

**Update**: Michael Harris is appearing via Skype from his home near here, since transportation out of NYC yesterday was mostly shut down (very early season unprecedented snowstorm, during rush hour…).

**Update**: I’m listening to the closing talk by Langlands. He is explaining his version of geometric Langlands, responds to criticism from Frenkel with “As far as I know there are no errors in the paper, no matter what you may see elsewhere”. He ends his talk with something like “At the last page I threw down my pen… It works and it works by a miracle. Don’t doubt it, it does work!”

**Update**: Another livestream, starting in moments: Alice and Bob Meet the Wall of Fire, a panel discussion with Quanta writers at the Simons Foundation.

**Update**: Videos from the Langlands Abel conference are now available, in particular Frenkel here and Langlands here.

**Update:** For another expository piece about the Langlands program, one that I somehow missed when it came out recently, see Sol Friedberg’s What is the Langlands Program? in the AMS Notices.

**Update**: An updated version of Frenkel’s notes is now available at the arXiv. Highly recommended for its lucid explanation of the form the geometric Langlands program has taken.

Langlands had expressed his scepticism regarding the “geometric Langlands program” already 2014 in Oxford at Symmetries and correspondences in number theory, geometry, algebra, physics. His talk was by video link, but the statements appear also in accompanying files that he had sent along, recorded here (search each of the three texts for “geometric”).

In the document called “message to Sarnak” he already said he is “uneasy about associating” his original conjectures with the physics S-duality-related ideas.

At the same meeting, a day or two after Langland’s talk, Edward Frenkel gave the usual exposition of Witten’s physics story advertisement for “geometric Langlands”. So I felt there must have been some communication gap.

It’s interesting that the article frames the issue as applied vs. pure, with geometric Langlands on the applied side. My impression is that number theorists are suspicious of geometric Langlands because it is so categorical, and it can be hard to tell how much there is there that’s “real.” This is one case where the “pure” mathematicians are less abstract than the “applied” ones.

Grad Student,

I agree that the article’s characterization of geometric Langlands as more “real”, or more “applied” is problematic. As practiced by Gaitsgory and others, the geometric version of Langlands is significantly more abstract (conceived of as a “categorical” statement) than the number field version. I would guess this is part of the problem Langlands has with it, that he was trying to do something more concrete. The force of Frenkel’s critique I think was to show that the concrete generalization that Langlands wanted couldn’t possibly work, you need to do something new, and that’s a reason behind the categorical formulation. His comments about recent work with Kazhdan were about an alternative more concrete approach, which might appeal more to Langlands than the categorical stuff.

Urs/Grad Student,

The other reason for Langlands to be uncomfortable is the recasting of the subject in the language of quantum field theory dualities. There the setting is very different than in mathematics and raises very different questions.

I’m listening to Langlands right now, and he’s addressing this very issue, saying his objections were two-fold: he wanted a legitimate mathematical theory, independent of the physics concepts, and he wanted legitimate eigenfunctions, not “eigensheaves”.

My impression is that Smith’s work is (or at least was) somewhat under a cloud, most notably as it is written in rather unclear style (even for a second-year grad student). I don’t think he was quite ready for what would happen after he stuck it up on arXiv.

From what I’ve heard, Sarnak has had various students look at it, all whom have given up at one stage or another. I think Lenstra had some people in Leiden trying to delve into it too, with inconclusive results. I don’t know about other study groups, but the topic is indeed of high interest. Supposedly, the analytic number theory part has some issues (the part I’d understand best personally), but I guess if nothing else you can assume GRH there (so says Sarnak, while I’ve heard Granville will vouch for a sufficient variant of the needed inputs).

With due respect to Melanie Wood’s comment, I think there *has* been a lot of interest in this, but that no one understands it all yet (Smith has tried his best to clarify various matters in his talks, but it’s still slow going). They have been some public champions of the result (Zhang, Wood, Elkies), but I don’t think they’ve been through the details. I don’t think any of them has (e.g) tried to write a survey article that illuminates the main ideas more clearly.

Also, the sentiment that the work “is contingent on BSD” is rather misleading, as this is merely the factoid that Goldfeld originally made his conjecture for analytic ranks rather than algebraic, while he could just have easily have done the latter (Smith plays up this BSD angle in his Introduction for some reason).

On the other hand, I’m definitely not in the loop (just largely reporting what has been told to me, for, as I say, it was of high interest in the summer conferences rumor mill), and for all I know, the details have been worked out by now. I’m actually kind of surprised Quanta ran with the story now (as opposed to any other time in the last year or more), so my suspicion is that some sufficiently distinguished person has decided it’s OK.

Glashow’s reply to Krauss is a gem. (Note to self: I should read Inference more often.)

A video of the Quanta panel discussion is available here (talk starts at 2:45:00): https://www.youtube.com/watch?v=tpPK1Fha0NA

A small correction: Alex is a fourth year student, not third.

Hello, Peter — As a frequent reader of your blog who came to the party for the string theory critique, but who keeps bumping up against Langlands, I’m wondering if there is a reference to which you (or perhaps one of your readers) could point a curious reader who is completely ignorant of who/what Langlands (as it it appears to be both a person and a proper noun referring to a field of study) is? Ideally, the reference might be at the level of someone (ahem) who is conversant enough with math through freshman/sophomore calculus (as taught ca. 1965), alas, though, not at all with abstract algebra or geometry.

grad,

Thanks, fixed.

Patrick Dennis,

A couple books about this I can recommend are Frenkel’s “Love and Math” and Ash and Gross’s “Fearless Symmetry”. Unfortunately, even for trained mathematicians, the mathematics involved here is as complex and not part of the usual curriculum as it is deep. For the connections to physics and quantum field theory, even more so…

Peter, I notice that your blog seems to take the Langland’s Program quite a bit more seriously than String Theory. Do you think the former has a greater chance of actually giving new “testable physics” than the latter? TIA.

Ricardo Jimenez,

They’re really too very different kinds of things. The Langlands Program is a hugely successful source of deep insights into mathematics, with very active on-going research making continual progress. The connections of all this to physics are fragmentary and highly speculative. I personally think that some day we’ll see fundamental new insights about quantum gauge theories, relevant to understanding the Standard Model, but at the moment what we know about is mainly just relevant to topological quantum field theories. So, the question of relevance to fundamental physics is an open one.

The string theory research program, on the other hand, has not at worked out at all as hoped. I’d claim that, as an idea about how to unify physics, it has simply failed and is in the process of being abandoned. There has been mathematics that has come out of it, but nothing of the depth and scope of the Langland program.

Peter,

You write, “They’re really too very different kinds of things. The Langlands Program is a hugely successful source of deep insights into mathematics, with very active on-going research making continual progress. The connections of all this to physics are fragmentary and highly speculative.”

It’s actually interesting, though: everything you say here about the Langlands program is literally true about string theory *as mathematics.* I sometimes wonder if string theorists have just wandered into a very successful speculative geometry program that doesn’t happen to have anything to do with physics…..

(You say at the end that the mathematical contributions of string theory are not of the same depth and scope as Langlands. I know more about the former than the latter, but from where I sit that’s not so clear, and I say that while realizing that the insights of Langlands have been extremely deep.)

Langlands’ conjectures are hard to prove, but just

statingthem is not as mysterious as it may seem from popularizations. The most useful review that is right to the actual point (undistracted by the conditio humana) is maybe still this one:Stephen Gelbart,

“An elementary introduction to the Langlands program”,

Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219

[doi:10.1090/S0273-0979-1984-15237-6]

The crux of the matter, extracted in a few paragraphs, is here.

S,

I don’t want discussion here to devolve to the usual arguments over string theory, my mistake to have allowed and responded to the above comment. “String theory” is now an ill-defined term, often being used to refer to QFT. I’ve often argued that it’s the QFT aspects that are mostly what is behind important mathematical contributions, but from bitter experience know that entering into the thickets of that argument is a waste of time.