Talk:Langlands program

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Langlands' original program was much wider than this in scope, regardless of what's left of it today. Also the first sentence is not correct in that 'group' is singular...?

Langlands also intended to have impact beyond mathematical proofs on mathematical practice, according to some sources, e.g. Simon Singh, that's reflected in the deleted text. AxelBoldt is again out of bounds.

Where is the statement by Langlands himself?

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Langlands' original program was much wider than this in scope, regardless of what's left of it today.

Please provide the relevant citation so that we can check this claim. AxelBoldt 23:09 Jan 24, 2003 (UTC)

I think this article needs a section on the geometric Langlands program. I don't quite know enough about it yet to write it myself, however. Etale 09:44, 29 January 2007 (UTC)

What this article really needs is more down-to-earth description of some of the original conjectures. It may not be accessible to high school students, but at least, there should be some desription (and yes, it's possible to accomplish!) that will make sense to general scientifically minded audience, not just people with Ph.D. in Langlands program, geometric or not. 129.15.11.123 02:34, 8 March 2007 (UTC)

I'm sympathetic with the last comment. Everyone who writes encyclopedia articles should try to make them as accessible as possible.

However, the authors should be cut some slack, especially in this case. Some things on the cutting edge of research simply can't be explained well to a general audience. To explain something well you need to understand it well, and in this case we don't — that's what puts it on the cutting edge. The Langlands Program is notorious for being hard to explain to non-experts. Perhaps only 1% or so of professional, university-employed, research mathematicians have a decent grasp of what it is. 158.109.1.15 (talk) 02:45, 2 February 2008 (UTC)

That could well be true, but psychology teaches us that when we've bit off more than we can chew, to step back and reframe from a larger perspective. I liken this to the whole "Not Even Wrong" debate concerning the string theory "landscape" program. The argument over much of the last century for funding physics at this level was that experiment informed theory and the end result ("The standard Model") was at least suggestive of potential application. Research into "The Landscape" departs from this original social contract. It might be that society wishes to continue funding this, or perhaps not. If society does continue to fund this, it won't be for the same reasons that governed physics for most of the last century. This needs to be debated, yet many involved in Landscape research seem to resent the intrusion. Presumably with the Langlands program, elite universities are funding very smart people to invest their energies in this. What would success, if achieved, look like? What would it accomplish for society? What would it accomplish within the discipline of mathematics? It might be that there is only at present a hint or a hope that some deep result materializes. Fair enough, but not immune to explanation, no matter how abstruse the theory itself. MaxEnt (talk) 10:06, 16 March 2008 (UTC)

The Langlands program basically sets out to parametrize an interesting and important subset of the set of representations of reductive algebraic groups (such as the general linear groups, unitary groups, E₈, etc) over arithmetically interesting rings (the adele ring of an algebraic number field or a global function field, or a local field), and to describe relations between these parametrizations arising from relations between groups. Success would be just that. Beyond the questions answered in representation theory, this would have a huge impact on number theory as the parametrizing set is formed of things that are pretty much Galois representations. A successful program would allow for translations between two different subjects in math which allows to use results from both. The Langlands program is not at all what string theory is. Mathematicians don't just do a bunch of research "in the Langlands program", rather they study things that are already interesting to them, but use the insight that Langlands had as hints at what should be true, and for what reason it should be true. Examples include Fermat's last theorem and the Sato-Tate conjecture and more, i.e. several deep results have already materialized. On a more constructive note, this article would probably be better if the conjecture was explained in terms of parametrizations by L-parameters. Though L-parameters aren't that transparent either, the statement of the program could be made more understandable. This is on a long list of mine. RobHar (talk) 18:53, 7 December 2008 (UTC)

I don't know if anyone is still following this thread, but I just wanted to say that I think the comparison to string theory is unfair, both to the Langlands program and to string theory. These are both well motivated research programs which have achieved many concrete successes. The "landscape" business that MaxEnt mentions accounts for hardly any of the current research in string theory, most of which deals with rather concrete mathematical problems in quantum field theory.

I also wanted to say that there are some very interesting connections between the Langlands program and mathematical physics which the article ought to mention. For example, Anton Kapustin and Edward Witten have shown that the geometric Langlands correspondence can be understood in terms of S-duality in four-dimensional supersymmetric gauge theory, and there are older connections between the Langlands program and two-dimensional conformal field theory. Both of these topics are closely related to string theory, by the way.

Since these topics are all related to the geometric Langlands program, I was thinking we should have a separate article on the geometric program. Does anyone know how to do that? At the moment, the phrase "Geometric Langlands correspondence" redirects to a section of this article… Polytope24 (talk) 05:43, 27 August 2013 (UTC)

Proof vs. conjecture - "Love and math" book/physics

I'm not expert, but should the article say proof[en] [somewhere]?

Note: Langlands conjectures now redirect to Langlands program, while the former term is still uses (as sometime in math even when proven).

I'm reading "Love and math"-book (almost finished), and while I do not understand all of it (but do understand implications for string theory, and that the math might be sound just not as physics..(?) while not what the book says, on the contrary).

Googling now (as I do not have the book with me I find "Langlands correspondence implies" but also "proof", that I do not understand..:

"To summarize and complete, it is important to remember that in the case of number fields the correspondence between Galois representations and certain automorphic representations (the ones which are algebraic at infinity in the case of number fields) is only a part of the Langlands program, essentially because it concerns just certain automorphic representations, and Langlands functoriality is about all of them. [..] We can also do this for other groups (unitary, orthogonal, symplectic) but that gives no new Galois representations so I don't dwell on it (though in the proof, we need to do the case of these groups before going to GLn)."[1] and [2], [3]

Hopefully, my recent edits (not in this page or it's talk) are correct, but if this is only a conjecture (or correspondence?) it might be premature. I would really like to know if something has been proven (I recall the book said that), and the implications (for at least physics – there might be none – only in math).

[Note: I also just recently finished Lee Smolin's Time Reborn (much more understandable, without being "proofs") that seems to be the way forward for physics and possibly "contradicting" (might be compatible with?) string theory - at least he's not a "proponent of it or multiverse interpretation of quantum mechanics. comp.arch (talk) 18:32, 18 August 2015 (UTC)

Example - for the non-nerd to understand?!

Would it be possible to add an example to the very start of this entry? TrolleF (talk) 06:46, 27 March 2018 (UTC)

Implications for nonspecialists and lay readers

Added a brief summary to some of the implications of proving the central langlands conjectures for specific cases. It appears to be essential to make clear the widespread potential and meaning this project has on mathematics and science, as it is a field seemingly unknown outside of professionals in its specialization. Especially so, since it is a somewhat unique framework of its kind, having connections to such seemingly distant subjects as number theory, and algebraic geometry (with further influences in many others; including algebraic topology and arithmetic dynamics). Providing a fairly unified construction for solutions of vast analytical power in automorphic forms. Please discuss and add citations as much as possible. The laglands program must be better understood; therefore known.12:38, 16 July 2021 (UTC)

Added a simplified description of the program for specialists and possibly diligent lay readers. This is an imperfect reduction of the langlands conjectures to more well defined objects. If one can improve upon it or add citations please do, as such a lead description is indispensable for newcommers; both professional and not.12:18, 18 July 2021 (UTC)

Implications for the broader world?

Perhaps the article should begin with a statement like, "This program is a useless exercise among academics that will have no visible impact on the world at large." (I haven't looked for a source to cite for that, but it shouldn't be too hard to find. Here's an example not specific to the Langlands program: "Useless research - an expensive waste of time?". The Guardian.)

If there is evidence that the above statement isn't true, then let's put some of that evidence into the article. So far, it's all about how proving that the frammitz is confomashalicious to the grotflocker will make somebody's mathematical career or get them some more government funding.

Will success in this program make understanding economics any easier? Will it help humanity tame fusion power? Will it explain the spread of disease? Will it let us build more stable social structures, feed the poor, house the homeless, explain how ecosystems form or break, reduce deliberate online deceit and manipulation, help to resolve any alleged "crisis" in society? Will it allow bad actors to cheat currently working forms of cryptography? Will it help make cars and trucks use 0.03% fewer resources? Or will it just be a masturbatory exercise for the sole pleasure of its worker bees? Please, somebody who actually understands this, explain ANY useful purpose for this in Simple English. Why would any government burn up tax money extracted by force from actual working stiffs to fund this? Gnuish (talk) 00:39, 17 January 2022 (UTC)

Please see WP:NOTFORUM. --JBL (talk) 00:58, 17 January 2022 (UTC)

Attempting to tame back any emotional recoiling, one may have from such a blatantly aggressively ignorant statement; truly no offence intended as I was once in such shoes, I will attempt to answer specifically and generally this recurrent irksome query. Seeing as your background is fairly obviously related to sociology, one must not expect mathematical proficiency to be understood. As such, the somewhat seemingly insane terminologies and endless lingo of purest most abstract mathematics, do indeed appear borderline. However, this argument can be made for absolutely every, single, one, of the core sciences. Indeed, it is suchly often used in empty political and psuedointellectual rhetoric. The ironic fact is, the very devices we are using to communicate over such an empty critique, are powered by exactly these "useless exercises".

To answer directly, yes the Langlands program specifically has much potential, eventually, to be an enormously powerful tool in the reconciliation of physics; and other STEM fields. At the moment its extreme abstract nature is a bit far from being intelligible; even to your ""average joe"" PhD theoretical physicist. With that being said, seeing as one wishes to lampoon inflamed replies upon fields which evidently are poorly understood, it seems your insecurity about the fear that something so beyond your comprehension may be real; is truly feeding the flame of such ignorance. Those are shoes familiar to myself as well. Which I might add, once one unveils these new universes of comprehension, thus become amazing discoveries as to new visions of existence.

For the general answer, should science and therefore its queen mathematics, be carried out simply for the sake of itself? If you require an answer to that question, then it is clear an understanding of science is utterly lacking. Absolutely, unequivocally, the best of science was ever made, in the minds of pure critical thoughts. Taking logic beyond its finitary human limits. Which, as someone who spent some time working in this field, I can assure this program very much is. Again, this is not to ridicule a stranger anonymously. Merely an attempt to illusrtrate an understanding that science requires from everyone. We do this to answer the most inconceivable questions of existence. And yes, the price is often obscure impermeable terminologisms. Hopefully, the takeaway here is to understand, that a framework of depth and breadth such as this, deserves to be known; if not only for the mere fact of its ambitious attempt. Very much akin to, in its time hilbert's program, which failed miserably; to the benefit of everybody.

If you wish to broaden or elucidate, why this makes any sense to any material field of study, I shall simply say several millennium prize problems are directly related to these mathematical objects. And if that does not scratch the itch, then I will add that prime numbers which are the most central object of study in pure math, are connected left right up down and center, to nearly every physical system in existence (including our precious useless economy).

P.S

It costs nearly zip to finance pure mathematicians. Just some pencils really...

PP.S

I am not currently an active researcher in this particular field, and do not necessarily posses a personal bias to its explication. Merely an avid lover of mathematics and pure reason in logic.

I strangely hope somebody qutoes this somewhere.

194.116.51.147 (talk)