### What is the Langlands Programme?

#### Posted by Tom Leinster

You probably know that the 2010 Fields Medals have been announced.

One of the four medallists,
Ngô Bảo Châu, works on the
Langlands
programme. Now, I know the Langlands programme is famous. In particular,
Laurent Lafforgue won a Fields medal for *his*
work on the Langlands programme in 2002, so there was a flurry of
publicity surrounding it then.

The thing is, I’ve never succeeded in understanding the slightest thing about it. It’s as if it’s got a hard, shiny shell—it resists all attempts at explanation, even when the person listening is a trained, interested, mathematician. But I’d like to believe that isn’t so.

I’ve heard numerous attempted explanations in the past, but as soon as
I run into a term like “automorphic form” or “reductive group” I’m
lost. (I can go and look *those* up, and I have, but I still
haven’t learned anything about the Langlands programme itself.) I’ve
tried reading the two explanations on the ICM page, but
one’s pitched too high and the other too low. I’ve tried the
Wikipedia page, but it jumps suddenly from sentences such as

It is a way of organizing number theoretic data in terms of analytic objects

to sentences such as

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters.

I’d love it if someone could explain ** just one thing**
about the Langlands
programme in terms I’ll understand. If you can do that, I’ll have
learned more about it from you than I ever have from anyone else.

To give an indication of where I’m at, here’s what I think I know about the Langlands programme:

- it’s a fantastically bold, fantastically sweeping set of ideas linking together many parts of mathematics
- it’s been the source of a lot of influential work
- it has something to do with representation theory, algebraic geometry and number theory.

I’m afraid that’s the sum total of my knowledge, and I’m not even sure that the last one’s quite right. To make matters worse, I know very little about representation theory, algebraic geometry or number theory. For example, that second Wikipedia sentence contains five terms whose definitions I don’t know.

I’ll be very happy if someone can tell me something I can understand. I don’t expect to understand much, but even a tiny bit would be an advance. Thanks!

## Re: What is the Langlands Programme?

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.