Before I continue with brief descriptions of the laudationes, let me mention that Julie Rehmeyer has written descriptions of their work for a general audience and Terence Tao has now posted about the work of the Fields medallists and the other prizewinners. And as I have already said, the ICM website has links to the full texts of the laudationes themselves. So anybody now wanting to understand the mathematics has an excellent starting point, and I am free to concentrate on the more frivolous details of the talks, perhaps slipping in the odd mathematical comment as I do so.

Jim Arthur went next. His was the terrifying task (though much less terrifying for him than for most) of explaining the work of Ngô Bảo Châu to a general mathematical audience. I’d say that he did about as well as it is possible to do, which meant that he was able to convey some of the flavour, but obviously without managing to transmit to the non-expert the sort of wisdom tht it takes the experts in this particular area years to accumulate.

Ngô’s big result is a proof of the so-called Fundamental Lemma, conjectured by Langlands. Not bad to get a Fields medal for a lemma, you might think. The story here is that Langlands, while working out his famous programme, recognised the need for this lemma, and thought that it would be reasonably straightforward to prove. However, it turned out that he had wildly underestimated its difficulty. As Arthur put it, what Langlands was seeing was the visible part of the iceberg, and to prove the lemma it was necessary to uncover and understand the iceberg in its entirety. (I can’t remember his exact words, but he did not use the phrase “tip of the iceberg” and definitely did talk about icebergs.)

Arthur then gave a general description of the Langlands programme (having apologized in advance that much of what he was going to say would be “murky”). After the laudationes I found myself having coffee with Assaf Naor and Irit Dinur. (If you haven’t heard of them, they are both fabulous mathematicians, and both speaking here. In fact, Irit has just given her talk, the theoretical computer science plenary lecture, about the famous PCP theorem, of which she has a famous new proof.) 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. I’m not sure why I’m musing on this at such length, but one more thought is that the question, “What is the most general statement of which Gauss’s law of quadratic reciprocity is a particular example?” is an obviously entirely valid and interesting one, and if I understand correctly, one of the Langlands conjectures is more or less a proposed answer to it.

I don’t understand what an automorphic form is, but there are levels of non-understanding (I would be enjoying several deeper ones later in the talk) and Jim Arthur lifted me to a slightly higher one — by which I mean that I had a slightly better idea what automorphic forms were after the next section of his talk. Before, I just thought of them as particularly nice kinds of functions that number theorists liked, and often mentioned in the same breath as modular forms (which I understand slightly better but still by no means fully). Anyhow, automorphic forms are eigenforms of natural operators on arithmetic symmetric spaces. He then said that these natural operators were Hecke operators, which themselves were Laplace-Beltrami operators on … er … I can’t remember. Hang on, we’ve got some spaces around — those symmetric spaces — so that’s OK.

What does one get out of a portion of talk like that? That is, what does one get out of hearing one concept one does not understand explained in terms of others? Let me try to say in this case. I don’t know exactly what an eigenform is but I presume it’s an eigenvector (and in fact he described them as simultaneous eigenvectors, so perhaps they were simultaneous eigenvectors for all Hecke operators — hmm, not sure about that). He talks about natural operators, which is obviously not meant to be precise and can therefore be understood in a non-precise way. Then he said “arithmetic symmetric spaces”. I don’t know what those are, though I imagine they are one of those definitions that is rather simple when you finally get told it. (I had that experience with algebraic groups, objects that I was afraid of until I learned that they were just groups where the set and the group operation are defined by means of polynomials.) I happen to know that a Laplace-Beltrami operator is what you get when you ask what the right analogue of the Laplace operator should be for a function defined on a manifold. (These last two definitions I know only as a result of editing the Princeton Companion to Mathematics, which forced me to pick up quite a lot of this kind of general knowledge.) And I’ve heard Hecke operators mentioned numerous times without ever actually finding out what they are. As a result of all that I now know that automorphic forms are eigenfunctions of operators that come up in a nice natural way in a number-theoretic context and that relate to all sorts of buzzwords I’ve heard many times. That doesn’t tell me exactly what an automorphic form is but it is non-trivial information. (What are they good for? There I cannot say anything that’s worth saying.)

Anyhow, the Langlands programme is about connecting automorphic forms with representation theory and looking at objects called automorphic representations. The difficulty of the area is this. One has some nice concrete operators (the Hecke operators mentioned above) and would like to know about their eigenvalues. However, just because an operator is concrete, it doesn’t mean you can write down its eigenvalues, and in this case you can’t. However, what you can hope to do is relate automorphic representations for different groups to each other, and this, if you manage it, gives you very deep reciprocity laws.

There’s something else called the principle of functoriality, which I won’t attempt to describe here even vaguely. Jim Arthur said, “The principle of functoriality awaits the efforts of future Fields medallists.”

The slides were getting more and more difficult to get anything out of by this stage, and I think I won’t say very much more. But Arthur gave us some idea of why the Fundamental Lemma has so many interesting consequences, and then started to explain what was so remarkable about Ngô’s proof.

If you want to impress your friends, here’s how to pretend you understand the proof in detail. If someone asks what his main idea was, you can reply, “Well, his deepest insight was to show that the Hitchin fibration of the anisotropic part of the trace formula is a Deligne-Mumford stack.” If that doesn’t do the job, then try to drop the phrase “perverse sheaves” into the conversation — they are relevant apparently. If you want to show that you have a broad view, then you could also say that Ngô very remarkably used global methods such as Hitchin fibrations to prove a local theorem. If you’re looking for a single amazing idea, then probably the use of Hitchin fibrations was it. Finally, here’s a list of names to splash about: Goresky, Hales, Kottwitz, Langlands, Laumon, MacPherson, Shelstad and Waldspurger. (This is apparently a far from complete list of the people on whose work Ngo builds.) A final summary from Jim Arthur: Ngô’s work opens up automorphic forms to some wonderful applications.

August 21, 2010 at 12:15 pm |

Dear Tim,

I recently gave what I hope was an accessible – and certainly very informal! – talk on the fundamental lemma – notes and video at http://media.cit.utexas.edu/math-grasp/David_Ben-Zvi_lecture.html

(a much better attempt by David Nadler will appear in the Bulletin of AMS I believe). It seems one needs to find amateurs like me who are willing to lie enough to make this more broadly comprehensible.. In any case, to summarize, Ngo has made a great breakthrough in our understanding of the relation between eigenvalues, characteristic polynomials and conjugacy classes of matrices, in a form suited to deepen the ancient theme of the relation between conjugacy classes and representations in groups… Hope that helps.

David

August 21, 2010 at 12:20 pm

I’ll definitely check that out. Many thanks for letting me (and others) know.

August 21, 2010 at 4:13 pm

I tried to watch the talk, but apparently my browser is not able to embed Quicktime video. Moreover, there is no link to download the file itself to view on a local video player. Would it be possible to either make the video available in another format or making the video available for download?

Thank you

August 21, 2010 at 4:33 pm

Never mind, I’m able both to watch it embedded and to download it using the version for iPhone. Maybe it is just a problem with embedding a video with Flash?

August 21, 2010 at 1:03 pm |

Highly recommended. It’s an awesome talk for aspiring mathematicians. Especially for those interested in that area -it’s a great introduction-, or in David. :)

August 21, 2010 at 2:22 pm |

One of the things I find really heartening about these and all your blog posts is that this is obviously

notwhat you’re setting out to do: impress your friends. The absolutely uninhibited intellectual honesty, not only in confessing ignorance and confusion (as in this post), but also thinking hard and out loud about ostensibly “elementary” topics on occasion. I cannot find a trace of glibness or braggadocio in anything you write, and that’s really appreciated. Wish more people would write this way!Hard for me to guess whether being a Fields Medalist makes it easier or harder. :-)

August 21, 2010 at 9:00 pm

Couldn’t agree more!

August 21, 2010 at 2:33 pm |

Regarding Hecke operators: on an arithmetic locally symmetric space, in addition to the notion of “nearby” coming from the Riemannian metric, there is a combinatorial notion of “nearby” coming from the number-theoretic symmetries of the manifold, in fact infinitely many such notions — each for one prime number.

In the simplest cases for all (but finitely many) prime numbers p there is a naturally defined graph (with vertex degrees depending on p) whose vertex set is the manifold. The “Hecke operator” is then the combinatorial graph Laplacian. Moreover, all the notions of “nearby” commute — locally speaking moving to a neighbour in the “p – graph structure” commutes with moving to a neighbour in the “q – graph structure” where p,q are different primes, and similarly moving a small distance in the Riemannian metric commutes with moving along one of the graphs. What this means is that all these Laplacians commute.

Since the Laplace-Beltrami operator and all these combinatorial Laplacians commute (and they are all self-adjoint) there are functions on the manifold which are joint eigenvectors of all of them, and these are the functions under consideration.

The structure of the graph you get is well-understood — except for a set of vertices of measure zero (in the usual sense on the manifold), the connected components are all trees, so the graph structure is basically a forest.

In greater generality what you get is not always a disjoint union of trees but usually a disjoint union of higher-dimensional simplicial complexes, but the basic idea is the same.

August 21, 2010 at 7:43 pm

Many thanks for that — the fog is clearing further as a result.

August 22, 2010 at 6:40 am |

I was wondering whether I would be the first person to be provoked into saying that Hecke operators *commute* with Laplace-Beltrami operators (though they can themselves be seen as a discrete or p-adic analogue of Laplace operators, roughly speaking). I think I am the second person.

August 22, 2010 at 7:13 am

Even though you are the second, it is only now that the message is sinking in.

August 22, 2010 at 6:53 am |

And modular forms are just the classical case of automorphic forms: modular forms are automorphic forms on the upper half plane (= the one-dimensional complex manifold consisting of complex numbers with positive real part, endowed with the usual Riemannian metric). To confuse matters slightly, people nowadays prefer to define automorphic forms on groups and their quotients; one then has to prove the not completely trivial fact that there is a simple transformation that gives you a bijection between modular forms (with certain parameters specified, e.g., the weight k) and automorphic forms on the group SL_2(R) (with the corresponding parameters specified accordingly – e.g., the same weight k).

August 22, 2010 at 7:16 am |

Unrelatedly, I would like to mention something a bit curious that I’ve noticed in the abstracts booklet located at http://www.icm2010.in/wp-content/icmfiles/abstracts/Contributed-Abstracts-5July2010.pdf

Indeed while very estimeed researchers make short communications in the number theory section, e.g. Granville/Alon/Ubis-Martinez at page 102 about sumsets in finite fields, there appears to be very possibly bogus contributions by amateurs: I’ve found no less than 4 different abstracts claiming a proof of FLT!! See pages 72-73, 86-87, 92-93 and 99-100. I’m a bit puzzled how these have been accepted at all, I hope they won’t appear in the proceedings. The same phenomenom is perhaps occuring in other sections, I haven’t had a look there yet.

August 22, 2010 at 1:43 pm |

To clarify, I asked the above question in the context of our discussion on how one could explain long-term research programs in a talk intended for the general mathematical audience. The research programs of this type with which I am closely familiar do not usually lend themselves to such an exercise, but I suspect that the Langlands program might. If it were possible to give an elementary number theory result that is currently unproven, but would follow from the assumption that all the Langlands conjectures are true, then it would be a cool way to start a general audience talk on this topic. This isn’t necessary, it is definitely not a criterion to judge an area, and I completely agree with you that in good mathematics often attention must turn to more global phenomena. I asked a couple of experts, and they didn’t know of immediate applications to Diophantine equations off the top of their head, but my hunch is that there should be such applications that are easily stated, and if so it seems worthwhile to work them out, if only for the purpose of talks.

August 22, 2010 at 5:48 pm

Thanks for that — sorry not to have consulted you before reporting on our conversation, but I hope that I didn’t misrepresent your views too much, even if what you say in your comment is more nuanced …

August 22, 2010 at 6:44 pm

Along these lines, one of my first courses on automorphic forms introduced (parts of) Langlands’s conjectures as follows: if you look, for some fixed k, at the arithmetic function r_k(n) which is the number of representations of n as sum of k squares (say k=24 for concreteness), then there is an asymptotic formula where the main term is an “elementary” sum-of-divisor-type function, and an error term. How large can this error term be as n grows? The Ramanujan-Petersson gives the answer, and the point is that this can be predicted very easily using the conjectured (by Langlands) existence of what are called “symmetric powers” of classical modular forms (it’s based on a variant of the “tensor power” trick that Tao blogged about a while ago).

Now, it turns out that — in that special case — the desired conclusion was proved without going through symmetric powers. But that was not easy either: Deligne did this using both his proof of the general Riemann Hypothesis over finite fields, and some highly non-trivial algebraic geometry. But his proof of RH over finite field uses itself some tensor-power trick, and I’ve heard it suggested that this was inspired by Langlands’s conjectural approach.

In any case there are many instances of more general cases of the Ramanujan-Petersson conjecture which remain unproved… but would follow immediately from general forms of Langlands functoriality.

P.S. Incidentally, Deligne’s result for the Ramanujan tau function, which is more or less the case of 24 squares, is on the logo of the ICM 2010…

August 22, 2010 at 8:36 pm |

Emmanuel: RH is an unbelievable problem because of the near misses. These other theories building around it: do they have a similar phenomenon?

August 24, 2010 at 9:43 pm |

Here is one consequence of Langlands’s conjectures: if $X$ is a quotient of the upper half-plane by a congruence subgroup of SL_2(Z) (i.e. by the kernel of the map

SL_2(Z) –> SL_2(Z/nZ) for some n), then the least eigenvalue of the Laplacian on L_2(X) is at least 1/4.

(This is Selberg’s 1/4 conjecture; it is an analogue for the Laplacian of the Ramanujan–Petersson conjecture mentioned by Emmanuel. It follows from the same tensor power technique that Emmanuel describes; indeed, in Langlands’s viewpoint, the Laplacians and the Hecke operators have exactly the same status, and functoriality for symmetric powers of GL_2 (which is currently still a conjecture!) implies both Ramanujan—Petersson and Selberg.)

August 31, 2010 at 2:13 pm |

I am a Vietnamese and I’ll help you add the accents: Ngô Bảo Châu.

August 31, 2010 at 2:27 pm

That was a great help — accents now added.

September 3, 2010 at 11:19 pm |

[...] ICM2010 – Ngo laudatio (gowers.wordpress.com) [...]

September 12, 2010 at 6:17 am |

In the course of this strange task I took on of explaining to a Korean journalist the significance of the fundamental lemma, I came upon this entry. Perhaps I can contribute one or two remarks on this rather complex topic.

As far as *direct applications* of the Langlands programme to Diophantine problems are concerned, they tend to be confined to the study of abelian varieties, that is, elliptic curves and their higher-dimensional generalizations. This derives from an approach to Diophantine equations that proceeds through the study of -functions. That is, to an equation , one can associate a function in an complex variable , that has the form of the Riemann zeta function in its Euler product form. It’s something of an article of faith that these functions encode deep arithmetic information about . There are a number of situations where this faith seems reasonably well-justified, such as cubic equations in two variables (elliptic curves), and made precise in the conjectures of Birch and Swinnerton-Dyer. In any case, regardless of how devout you are, an embarrassing fact is that is in general defined as a horrible infinite product over primes, and it’s not clear that it’s a reasonable function of any sort at all. Another article of faith, which comes in some sense before the previous one. is that this function is indeed nice: It should have a continuation to the whole plane, and satisfy a natural functional equation. One of the main points of the Langlands programme is to prove this statement through an identification

$$

where is one of these things called an automorphic form. [Actually, it should in general be a more complicated object called an automorphic representation of a large group, $GL_n$ with entries in the adeles of an algebraic number field. A rather simple way to show sophistication in this subject, even without going into all the buzzwords mentioned at the end of the post, is to casually use the two terms `automorphic form' and `automorphic representation' interchangeably.] Because $L(f,s)$ is associated to a topological group, harmonic analysis has been used to show already all the nice analytic properties one could hope for.

Anyways, the point I am trying to make is that the Langlands programme does propose to contribute to a deep and systematic study of *all* equations. Unfortunately, the kind of information contained in these tend to of the *abelianized* sort, something like the free-abelian group generated by the solutions modulo some geometric relations. The reason there are direct implications in the case of abelian varieties is because the naive solutions then form a group, so that the abelianized information and the `direct’ information coincide. It is a somewhat speculative but serious challenge for arithmeticians to come up with a non-abelian version of this whole picture, whereby one might come to a systematic (albeit conjectural) understanding of naive solution sets for rather general equations in a manner resembling the abelian situation.

Meanwhile, I might remark also that one can always try to use the understanding of elliptic curves or abelian varieties gained through -functions to study other equations, for example, by getting solutions of an equation to *parametrize* elliptic curves or abelian varieties. This is what happened in the theorems of Wiles and of Faltings.

My feeling is this old ICM lecture of Langlands paints a good picture of the subject in very broad brushstrokes. Perhaps one sentence there worth retaining is

`However, all evidence indicates there are fewer -functions than the definitions suggest, and that every -function, motivic or automorphic, is equal to a standard function.

The mysterious term `motivic’ -function refers (essentially) to one associated to a Diophantine equation, while a standard - function is one coming from an automorphic represention of .

Finally, I believe the fundamental lemma itself has direct application to a version of the -adic analogue of the BSD conjecture, as studied by Chris Skinner and Eric Urban. [I'm not entirely sure about this though, and I hope some real expert will confirm or deny this.]

September 12, 2010 at 6:26 am |

One more remark: as implied in the Langlands quote, there are -functions associated to automorphic representations of more general groups, such as unitary groups or symplectic groups. However, Langlands is saying that as far as -functions are concerned, we should always be able to reduce to the case, where everything is relatively simple. Ngo’s theorem implies some special but important cases of this last statement (and more).

September 14, 2010 at 5:22 pm |

[...] ICM2010 – Ngo laudatio (gowers.wordpress.com) [...]

October 15, 2011 at 7:51 pm |

[...] certifiably one of the best mathematicians in the world. Let me grab a couple of sentences from him (here on Tim Gowers’ blog):. I don’t understand what an automorphic form is, but there are levels of non-understanding (I [...]

December 12, 2012 at 5:50 am |

[...] Gowers main point is that a significant part of the mathematical establishment looks down on combinatorics as not being “deep”, and that this attitude is both harmful and ignorant. On this point, I think that Gowers criticisms are fair, accurate, and valuable. It’s undeniably true that there are many graduate students who fall in love with formalism to the detriment of content, and milder forms of this predujice are pervasive throughout mathematics. To this end, I think Gowers’ essay is timely and relevant. However, I can’t help but sense a little that, perhaps after having spent a career defending combinatorics against ignorant snobs, Gowers suffers from the opposite prejudice, where “theory-builders” are a short distance away from empty formalists, sitting comfortably in their armchairs thinking deep thoughts, studying questions so self referential that they no longer have any application to the original questions which motivated them (this sense also comes from reading some of the remarks on the Langlands programme here). [...]

March 21, 2013 at 3:22 pm |

[...] for the mathematics community as a whole. My feelings on this subject were reinforced in 2010 when experts tried in vain to explain the work of Fields Medalist Ngô Bảu Châu to the mathematical [...]