The n-Category Café

http://online.itp.ucsb.edu/online/duallang-m10/

You could begin by thinking categorically. The so called categorical Langlands conjecture says: the category of some-modules for Bun(G) is equivalent to the category of some-modules for Loc(G*) where G and G* are dual Langlands groups, and Bun(G) means bundles for some curve C, and Loc stands for ‘local systems’ for some curve C. See the above online talks.

So even the ‘categorical’ version involves tons of classical geometry, and is not very categorical. The most categorical part seems to be the links to Chern-Simons and Khovanov homology and other knotty theories, despite Kapranov’s attempt some years ago to talk about 2-cat structures associated to Langlands. For me, also at your level, the trickiest thing is sorting out the different kinds of Langlands: classical, geometric, bla bla.

Maybe this will help, I’ll go through and explain what these things are in GEOMETRIC Langlands (and over the complex numbers), which works by analogy with the arithmetic version.

dual Langlands groups: Because in complex geometry we’re using reductive Lie groups, we can just use real compact Lie groups, and then complexify. Now, with real compact Lie groups, there’s a poset, with simply connected and centerless (adjoint form) Lie groups at the ends, consisting of all Lie groups with the same Lie algebra. The Langlands dual is what you get if you switch the order of the poset around, and then match things up.

local system: a G-local system principal G-bundle with flat connection

curve: for the purposes of my response, literally just a Riemann surface.

bundle for a curve: principal G-bundle

module for Bun(G): Here’s the complication. Bun(G) is a stack, the moduli stack of principal G-bundles on the curve C. And Loc(G) is also a stack, the moduli stack of G-local systems on C.

So the geometric Langlands correspondence would be an equivalence between local systems for the Langlands dual of G on a curve and eigensheaves for a “Hecke” operator on the stack Bun(G).

I just found this paper of Witten, which seems to do a decent job explaining things (I only skimmed) and ties Langlands duality into electro-magnetic duality: http://front.math.ucdavis.edu/0906.2747

I succeeded in understanding one small thing about the Langlands programme, from Ian Grojnowski’s article on representation theory in the Princeton Companion to Mathematics. He says that

The Langlands program describes the representation theory of reductive groups.

(That’s not quite a direct quotation.) Now, I didn’t and still don’t know what a reductive group is, but I do know now that they’re algebraic groups, and that many important algebraic groups are reductive. So, I can understand the following statement:

The Langlands program describes the representation theory of many important algebraic groups.

That was something I didn’t know.

Grojnowski indicates that, according to Langlands, the representations of a reductive group are described in terms of (a) the Langlands dual group and (b) some Galois group. I guess (b) has something to do with the connections with number theory.

However, I’m still missing a sense of why the Langlands programme has such a broad sweep. A couple of paragraphs later, Grojnowski says:

It precisely unifies and generalizes harmonic analysis and number theory.

How does harmonic analysis come into it?

And… that’s an enormous claim. What does this unified, generalized, harmonic-analysis-number-theory look like? Is that what all these black-belt Langlands dudes are busy finding out?

Here’s my submission, which may be redundant.

There’s a special class of groups, called connected reductive groups, that are parameterized by combinatorial data (root data). This combinatorics is pretty simple, not too much more complicated than the classification of Lie algebras by graphs.

(I am trying not to burden anyone with any information about what a reductive group is. I hope this doesn’t count: reductive groups are groups of matrices, like GL(n) or SO(n). In three paragraphs I will say something about what kind of numbers to fill the matrices with.)

Grothendieck observed that root data come in dual pairs. At the time it might have looked only like a curiosity: without explaining anything, I can tell you that a root datum is a four-tuple (X,Y,R,C), and it turns out that (Y,X,C,R) is another root datum.

Since root data come in dual pairs, reductive groups also come in dual pairs. The partner G’ of a reductive group G is called its “Langlands dual.” There are by now startlingly intricate geometric explanations of how to build G’ directly out of G, like in Allen’s comments.

The Langlands program starts (and I will almost stop) with the following claim: homomorphisms into G (Galois representations) should be related to homomorphisms out of G’ (automorphic forms).

Actually the claim should have some number theory in it: the short story I told about G and G’ was basically true over the complex numbers (or another algebraically closed field), but in the Langlands program a number field F is involved. F plays a different role for G than it does for G’. A Galois representation is a homomorphism from Gal(F) to G-with-complex-entries. But an automorphic representation is an action of G’-with-adeles-of-F-entries on a topological vector space. (That’s where the harmonic analysis is.)

(When F is the rational numbers, an “adele” is a sequence [real number, 2-adic number, 3-adic number, 5-adic number, …] satisfying a condition.)

One sentence to try to connect this with what Charles is saying about geometric Langlands: using the dictionary between number fields and function fields, Weil suggested that G’-with-adelic-entries is analogous to the group of gauge transformations of a principal G’-bundle over a Riemann surface.

Try listening to David Ben-Zvi in this lecture on the Fundamental Lemma.

Gowers’s Weblog has a link to a well-written (and simple– I hope not too simple) summary of the Langlands program that also discusses Chau’s work.

ICM2010 — Ngo laudatio

“… The conversation turned to the topic of how many Fields medals could in theory be given for advances in the Langlands programme. My view was I suppose the official line, which is that it is such a deep and difficult area that any major advance is huge news, though I couldn’t resist a joke comparison to pole vault records, where people who are in a position to beat them deliberately don’t beat them by much because you get big money for beating world records. (I’m not seriously suggesting that somebody who had a proof of all the Langlands conjectures would sit on it, or release it only gradually.) Assaf (and I hope he won’t mind my making his views public) was more sceptical, maintaining that all mathematicians have their icebergs to explore and that the Langlands programme was not as unusual in this respect as perhaps it is sometimes conveyed as being. He said that he likes to ask the experts whether if they could assume all the results they wanted that are currently conjectural, they would know more about any concrete Diophantine equations. Apparently they don’t particularly like this question. Whether it is an appropriate criterion to judge the area is of course a matter for debate. In fact, that is what prompted me to say that perhaps the iceberg was the true and fascinating object of study in that area. I didn’t think of saying it at the time, but after a while there is not much interest in solving more and more Diophantine equations (not that Assaf was claiming that there was), and attention must turn to more global phenomena somehow. Perhaps that is what algebraic number theory is….”

As Peter points out it’s useful to separate the local and global parts of the story. The local Langlands program is a general principle that tells us what representations of groups like GLn, SLn, SOn, Spn (all your favorite matrix groups) look like, if you take the entries of the matrices to be real, complex or p-adic numbers (though the same principles work also for finite fields and other settings). Namely there’s a notion of “nice” representation – these include all finite dim reps but also all reasonable infinite dimensional ones…

The local Langlands program says these representations are classified by “Langlands parameters”. What are these? remember for a finite group there is the same number of conjugacy classes and irreducible reps, but no canonical way to match them up. The idea very roughly is that reps should match CANONICALLY with conjugacy classes in ANOTHER group, the dual group. Something like this is the content of the Langlands program over finite fields (due to Lusztig). For the other fields it’s still generally the idea, except rather than looking at conjugacy classes of ELEMENTS (aka homomorphisms from Z) in the dual group, we look at conjugacy classes of homomorphisms from a Galois group attached to the situation. Depending on the field then you may decide where the complexity lies — for p-adic groups the Galois group is super intricate, while over R or C it’s easy, so in the former you think of this really as Galois data while in the latter you think of it basically as conjugacy classes in this dual group (for which there’s always an easy combinatorical prescription). That’s it in a nutshell.

Then there’s the global Langlands program which says what happens when you look at fields like rational (meromorphic) functions on a curve (Riemann surface), or more importantly, number fields. In this case we don’t describe ALL representations of our matrix groups, but the “deep” ones, aka automorphic. These are by definition the representations that arise when we study harmonic analysis on special locally symmetric spaces, like the modular curve (upper half plane mod SL2Z). Again the classification of these is by conjugacy classes of reps of a Galois group, which is now a super hard and subtle Galois group (eg that of the rational numbers). This correspondence of reps and Galois data ends up being the most powerful tool we have to understand these Galois groups, hence a huge variety of problems in number theory (such as Fermat’s last theorem etc etc…)

You may find it helpful to have a look at the paper “An elementary introduction to the Langlands program”, by Stephen Gelbart (AMS Bulletin, vol.10, number 2, 1984). It is very clearly written and is aimed at non-experts.

This might be more your style: a recent Kapustin paper on TQFTs.

If I were to recommend just one article that attempts to provide a large-scale view of the program, it would be Langlands’ lecture at the Helsinki congress.

In brief, he explains:

(1) The arithmetic of varieties are expected to be reflected in special values of their $L$-functions, or natural factors of their $L$-functions (that is, motivic $L$-functions), as in the class number formula and the conjecture of Birch and Swinnerton-Dyer (generalized by Deligne, Bloch, Beilinson, and Kato over the years subsequent to this lecture).

(2) The special values in question often lie outside the domains of convergence of the initial definitions as Euler products, and hence, call for analytic continuation, as in the conjecture of Hasse and Weil. (In B-S-D, one must take the value at 1, where the original Euler product doesn’t converge.)

(3) All known cases of this analytic continuation rely on the identification of the arithmetic-geometric $L$-function with an automorphic $L$-function, i.e., an $L$-function associated to an automorphic representation of the adelic points $G(A_F)$ of a reductive group $G$. These usually have nice properties coming from harmonic analysis on the group.

These considerations leads to the conjecture

*Every motivic $L$-function is an automorphic $L$-function.*

Whether or not the words therein makes sense at yet, this sentence is really the core of the Langlands program. In a real sense, the rest are details. I know I’m not explaining anything in this post. But it might still be useful to take home that sentence as something to mull over and keep coming back to, at least if you’re serious about understanding Langlands.

By the way, even though it’s important at some point, for the big picture, you lose nothing by assuming $G$ to be $GL_n$. In fact, the so-called functoriality conjecture implies that we can always take $G$ to be $GL_n$. In short, don’t worry too much at the outset about the dual of a group.

“*Every motivic L-function is an automorphic L-function.*”

So, whatever is the meaning of it, if this conjecture is proven, the Langlands program will be finished. Is that it?

In light of Minhyong’s very lucid answers, it seems worth mentioning (especially in the context of Ngo’s work; and it seems that in the West he is referred to, Western-style, as Ngo, rather than as Chau) that there are two aspects to what is usually called “the Langlands program”.

The one that Minhyong explained, to the effect that motivic $L$-functions should also be automorphic $L$-functions, is known as reciprocity. Roughly, one thinks of “motivic” as meaning “Diophantine”, so motivic $L$-functions encode Diophantine information (often in elaborate, and conjectural, ways, as in the Birch–Swinnerton-Dyer conjecture and its generalizations), while the automorphic $L$-functions are supposed to be thought of as accessible, understandable, perhaps even computable in some way.

A simple illustration of this kind of phenomenon is given by the famous result, due essentially to Gauss, that the Galois group of $\mathbf Q(\zeta_n)$ over $\mathbf Q$ (here $\zeta_n$ is a primitive $n$th root of 1), is canonically isomorphic to $(\mathbf Z/n)^{\times}$, where an element $a$ in this group of units acts on $\zeta_n$ just by raising to the $a$th power. The group $(\mathbf Z/n)^{\times}$ is a very simple example of an automorphic object (the underlying reductive group is just $GL_1$), and it is eminently understandable. With this isomorphism in hand, it is straightforward to solve the Diophantine problem of describing all subfields of $\mathbf Q(\zeta_n)$. For example, taking $n = p$, an odd prime, it is easy to deduce that $\mathbf Q(\zeta_p)$ has a unique quadratic subfield, equal to $\mathbf Q(\sqrt{\pm p})$ (the sign being chosen so that $\pm p \equiv 1$ mod 4), and using the description of Gal($\mathbf Q(\zeta_p)/\mathbf Q$) afforded in this way, one can easily deduce (for example) the law of quadratic reciprocity. (This is the origin of the appelation reciprocity for the general problem of relating motivic and automorphic $L$-functions.)

There is another aspect of the Langlands programme, though, in fact the one that Langlands first introduced, and in which the role of reductive groups other than $GL_n$ becomes more apparent, and this is the problem of functoriality . Very roughly, it says that if we have a map of dual groups $H^{\vee} \to G^{\vee}$, then automorphic forms on $H$ give rise to automorphic forms on $G$. This is where harmonic analysis enters the picture: describing/computing/constructing automorphic forms is a non-abelian analogue of Fourier theory (non-abelian because the groups $G$ and $H$ are typically non-abelian), and this problem is about constructing an automorphic form on $H$, given some corresponding data related to $G$.

Ngo’s theorem, the so-called Fundamental Lemma, allows one (through combining his work with earlier work of Langlands, Arthur, and many others) to solve this problem of functoriality in special cases, namely when $H$ is a so-called endoscopic group attached to $G$. (This is a very technical notion; roughly, it means that the map $H^{\vee} \to G^{\vee}$ is obtained by identifying $H^{\vee}$ with the centralizer of some element in $G^{\vee}$.)

Most instances of functoriality are not endoscopic, but nevertheless, the fact that even the endoscopic case could not be solved was a big obstruction (psychologically, as well as technically) to thinking about the general problem, so the fact that it is now essentially solved, thanks to Ngo, represents enormous progress.

Although functoriality and reciprocity are different problems, they are related heuristically and technically, and share many common features (often at a deep level, where it is not obvious that they should). Thus Ngo’s result also has implications in the theory of Shimura varieties, which is one of the basic testing grounds for reciprocity. Just to give an ideal of what I’m talking about here: Shimura varieties are certain varieties which are defined in terms of reductive groups, so that they are naturally quite close to the automorphic world; but nevertheless, proving reciprocity for them is a major task, not yet complete, and Ngo’s result has played a major role in recent advances on this front. (Let me close with a parenthetical remark directed at those readers who may know some of the theory of modular forms and modular curves, but don’t know the general theory of Shimuva varieties: as an illustration of the way the harmonic analysis problems related to functoriality and the Diophantine problems related to reciprocity intertwine, let me mention that Langlands discovered the phenomenon and problems of endoscopy in the context of trying to generalize Eichler–Shimura theory — the theory that relates cohomology of modular curves to modular forms — to the context of general Shimura varieties.)

This might be a good time to advertise this Workshop on Non-abelian Class Field Theory. It’s a small event with David Ben-Zvi, John Coates, Matt Emerton, and Bao Chau Ngo as speakers. The plans for this workshop arose from my earlier threat to bombard David with questions.

Availability of funds for participants other than the speakers is somewhat uncertain. But limited support for the local expenses might be possible. Let me know if you’re seriously interested.

So Tom, the guy who never installed the gothic fonts on his computer because he didn’t want to learn about Lie algebras, suddenly wants to learn about the Langlands program!

This gives me newfound courage. I’ve always been afraid of heights. I think tomorrow I’ll climb Mount Everest.

Being a complete novice at math(basic High school Algebra/Geometry)the Langlands program sounds like a cookbook that uses different math disciplines to approach various types of problems the answers to which themselves become part of the recipe combinations used to solve additional mathematics problems. The recipes/recipe combos themselves can become so complicated that THEY need to be solved before being used for “cooking up the solutions” to additional problems. I hope that my description is somewhat on the mark however as I’ve said my qualifications never went beyond the high school level.

Having recently received an email from John, I suddenly remembered something that was on my mind from years ago. That is, I had wanted to write something more coherent in this thread in regard to the question ‘why would anyone expect anything like Langlands functoriality’. Well, I gave a talk on this topic this summer. It’s not very good, but the goal was to answer exactly this question. So I’m posting a link here, for better or worse:

http://people.maths.ox.ac.uk/kimm/lectures/functoriality.pdf

For newcomers: An intriguing elementary entre is the Abel Prize Lecture by Frenkel on the Langlands Program.