# The Fundamental Lemma

So, Noah came up to New England a few days ago, and at some point over dinner, the topic of Fields Medal candidates came up. Neither of us had any good ideas (sorry, anonymous grad student) but I mentioned that I had heard Bao Châu Ngô’s name quite a bit. The conversation then went roughly like this:

Noah: Oh, really. What has he done?

Me: I think he proved the Fundamental Lemma.

Noah: What’s that?

Me: Ummmm…something to do with Langlands?….I’m not really sure.

Today, while Wikipedia surfing, I discovered that there is actually a document (I hesitate to call it a preprint) on the arXiv, entitled “A Statement of the Fundamental Lemma” by Thomas C. Hales. It is 18 pages long. Suddenly, I don’t feel so bad about not being able to state it off the top of my head.

Incidentally, for those of you wondering why someone would get a Fields Medal for proving a lemma, Hales explains:

There have been serious efforts over the past twenty years to prove the fundamental lemma. These efforts have not yet led to a proof. Thus, the fundamental lemma is not a lemma; it is a conjecture with a misleading name. Its name leads one to speculate that the authors of the conjecture may have severely underestimated the difficulty of the conjecture.

By the way, those of you who think this post has no point can consider this an invitation to a “summarize the Fundamental Lemma as concisely as possible” contest.  That, or you could make wild speculations about Fields Medalists.  I wonder which of those will happen.

## 36 thoughts on “The Fundamental Lemma”

1. If I may permit myself to comment about the Fundamental Lemma as I understand it: One tool in studying automorphic forms with a particular view towards attacking the Langlands conjectures is the trace formula. This is a particular identity with a spectral term on one side and a geometric term involving ‘orbital integrals’ on the other.

If one wants to get as much mileage as possible out of the trace formula, then one wants to be able to compare and identify these orbital integrals for different groups. This is where the fundamental lemma comes in, by providing these identities. Such identities turn out to be unexpectedly difficult to prove, hence the immense praise for Ngo’s work.

I will add another reference about the fundamental Lemma to those mentioned above, namely some notes of Waldspurger available at http://www.claymath.org/researchconference/2009/waldspurgerln.pdf

These are notes from a talk he gave at the Clay Institute earlier this year, at which I found that I got the greatest understanding out of all talks on the fundamental lemma that I’ve seen. It also wasn’t the first time I’d seen some of these ideas presented, which helped, since I don’t think it is realistic for most people to absorb the whole story of the fundamental lemma on a first pass.

2. I’m pretty sure that the lemmas Zeilberger is thinking about do not require 197 page proofs…

Ngo is definitely a strong candidate. What about Khovanov, Venkatesh, A.Kuznetsov, Artur Avila?

It is unlikely now that some one will come up with a solution of one of the millennium problems ( or something of similar fame) .
The EMS prizes are a good starting point though.

4. Know nothing says:

Someone has to be rewarded for Birkar-Cascini-Hacon-McKernan.

5. Someone has to be rewarded for Birkar-Cascini-Hacon-McKernan.

But do they have to be rewarded with a Field Medal? I’m very curious about the dynamics of how collaborations affect people’s Fields Medal chances. You can have someone like Terry Tao, who collaborates with tons of people, or Perelman, who did his most impressive work mostly on his own. I do feel like, however, someone like Ozsvath or Szabo would be seen as a much stronger contender if they hadn’t written most of their papers jointly (after all, unless only one of them is young enough, how can you choose just one?). Presumably there are similar examples in other fields.

6. math_lambda says:

I guess all depends on the very problem solved jointly: if two unknown folks prove the Riemann Hypothesis today with hugely new tools, I wouldn’t bet against both getting a Fields.

In other disciplines, that’s what happens for Nobels in physics I think, there are nearly always two or even three collaborators rewarded due to the way they work together. Even more so in biology/medicine: for example the 1985 prize on Cholesterol Metabolism was won by two close collaborators. These are disciplines where a team is the norm.

That said, it’s actually great that folks like Ozsvath and Szabo can adopt a “le’ts get a good joint result and come what may” philosophy even if it harms their chances, rather than become ultra-competitive.

Ben,
What about the collaborative work or Wendelin Werner and Oded Schramm.
In case of a major collaborative work, it is always tricky to attribute people, unless off-course age becomes a factor.

About Birkar-Cascini-Hacon-McKernan, I think their work is important , I just don’t know if it really fields medal important , or more importantly what are the other contenders?

8. math_lambda-

I think this is one of the most important differences between the Fields and Nobel prizes. Fields medals are never (can’t be?) awarded jointly, whereas Nobels are quite often. The Abel Prize (which is intended to be a closer analogue to the Nobel) has been jointly awarded twice, to Atiyah/Singer and Thompson/Tits. Notably in both pairs one is a Fields Laureate and the other not.

Of course, I don’t know the psychology of the Fields committee all that well, but I get the sense that for a single problem to be worth two Fields medals it would have to be on the level of the Riemann Hypothesis.

9. Antonis says:

There are some rumors in central Europe that Cedric Villani is also a strong candidate for a Fields Medal. He has done a lot of work on mathmetical physics (Boltzmann equation, etc) and in optimal transportation. Incidentally, we was also a recipient of an award at the EMS 2008.

10. Kevin says:

11. TheExige260Sport says:

Ngo Bao Chau is the best candidate, and he will get the fields medal this year.

12. As a footnote to this discussion, Ngo has accepted a
professorship at the University of Chicago (which did
not wait around for the Fields Committee).
Remarkably, Ngo’s adviser Gerard Laumon supervised
successively two superstars at Orsay: Laurent
Lafforgue and Ngo Bao Chau. That may be some kind
of world record.

13. antonio carlos motta says:

i think what the work of ben green is wonderfully.hacon is under 40?
chern medal,perhaps simon donaldon,for me the great mathematics,since 1980

14. Arctorius says:

It is clear that Ngo will get it. Isn’t Lindenstrauss just about young enough? He is an example of somebody very strong who has collaborated with plenty of people, but not the same ones each time.

My feeling is that Ben Green is not thought of as a serious candidate by most senior mathematicians, though some would append “unless politics works out very strangely”. Green and Tao’s theorem on the primes is very nice, but in some sense it is not about the primes at all, and does not tell us something deep peculiar to them. Rather, Szemeredi’s theorem tells us that any positive-density subset of the integers has arbitrarily long arithmetic progressions, and we know from sieve theory that the primes are a positive-density subset of (essentially) a set that is very much like the integers (or, more precisely, a sieve majorisation); G and T show (basically) that Gowers’ proof of Szemeredi is robust enough to adapt when the integers are replaced by a sieve majorisation.

Of course, the proof has played an important cultural role, in that it has made many number theorists and ergodic theorists share ideas.

15. Arctorius says:

See also: Avila, Bhargava, Lurie, Mirzakhani, Soundararajan, Venkatesh, though Lurie, Mirzakhani and Venkatesh have plenty of time.

16. Dear Arctorius,

I have heard remarks from very serious mathematicians that Ben Green is a candidate. The medal would not be for the original Green–Tao theorem (which you are referring to), but for the subsequent work of Green, Tao, and (most recently) Ziegler on the Hardy–Littlewood conjectures regarding linear equations in primes.
This work is spectacular, and is entirely about primes.

17. Lambert says:

Ngo should be a 100%. I don’t know enough about Villiani (the prize list is impressive though), but he seems likely (and Lions has influence I do imagine). I’d guess Hacon and company will be passed over, though if one person had proved the result, it would be in the “maybe” category.

Lurie is in a “Fields”-field research-wise, but I think it is too early.

Bhargava’s latest on Selmer groups won’t be soon enough, but he keeps getting mentioned. My personal guess he will never win (like many other NTers). Venkatesh seems more of an all-arounder than hitting one noted result (subconvexity can’t be that big?), and the Fields committees don’t often give much for that. He has time of course.

Lindenstrauss maybe (if Bourgain has sway), and Green too, particularly if you like the cross-fertilization with ergodic theory. The “prime” angle on Green’s results is a bit dodgy, and the word “spectacular” needs to be couched carefully, as they miss the interesting cases such as twin primes, and almost everything is conjectural. Again I think the “cross-fertilization” aspect of it can be played up (some would prefer to downplay there, though). The U^4 norm work with Ziegler does make it a bit less reliant on mere conjectures, though you could have the Ribet-Wiles problem there upon adding Ziegler as a co-author (as in, Ribet proved FLT “up to a technical hypothesis” as it were, and if modularity were known in the 80s, Ribet would be the name). One question is whether the Green-Tao paper on linear equations is considered to be the “technical” part, or the part with Ziegler gets that moniker.

Soundarajan is similar to Bhargava in breadth versus depth, and quantum unique ergodicity is not as important to physicists as was represented, which takes out a “main-result” consideration. I can’t really pick Mirzakhani at this point. Kedlaya I have heard mentioned, but I would think Bhargava over him.

Ávila is possible (also he is only 30), and a much more likely “we need another recipient” than any of the above.

Just to fill out names: Igor Rodnianski, Simon Brendle, Assof Naor, Ian Agol, …

So: Ngo 100%, Villiani 75%+, Ávila 50%, others less, led perhaps by Bhargava (think India).

The real question is who is on the Fields committee (Lovász is the chair, though it is unclear whether for “politics” with the IMU or for maths — he has chaired other committees, and my guess is that he won’t actually sway that much).

18. Lambert says:

I saw Hales talk in 2001 about the FL, but all I got out of it was the “why is it called a lemma if it’s the main conjecture” question. Now perhaps I understand it more.

19. Dear Lambert,

I don’t think it’s fair to say that Green and Tao miss “all the interesting cases” because they miss twin primes (and similar statements). That may indeed be a very interesting case, but the precise asymptotics of primes in arithmetic progression is pretty substantial too. (And of course there are many other interesting problems to which there results do apply.)

Also, why do you say it is based on conjectures? The Inverse Conjecture now seems to be proved (with Ziegler); the U^4 result has been publically released, and the general result announced. Whether this is enough, and in time enough, for the Fields committee, I certainly don’t know; but given the Green–Tao track record, there doesn’t seem any reason to doubt that they have proved what they claim to have proved.

As for the word “spectacular”, in the end I am speaking for myself: I find it so!

20. Dear Lambert,

One more comment/question: Why do you say that ” the “prime” angle … is a bit dodgy”? When I wrote that this work is entirely about primes, I had two things in mind: (a) it is about the precise Hardy–Littlewook asymptotics for linear equations in primes, so in a direct sense it is entirely about preims; (b) the (non)correlation result for the Mobius function and nil-sequences seems to be squarely in the tradition of classical analytic number theory, and so in a technical sense is “entirely about primes”.

The Inverse Conjecture, I guess, is not about primes, so perhaps “entirely” was too strong a word. (Although I don’t think that this reflects badly on the work; it is a fruitful combination of analytic number theory — the Mobius result — with additive combinatorics — the Inverse Conjecture.) However, I was reacting in part to the suggestion of Arctorius that this work is not really about primes at all, which is just wrong.

21. Typos: Littlewook |–> Littlewood

preims |–> primes

22. Lambert says:

Mikhail Khovanov might have a better chance than his publication list, as his work is seminal, and his field is not overloaded.

23. Lambert says:

I guess I can compare to Ford’s work ( http://en.wikipedia.org/wiki/Carmichael%27s_totient_function_conjecture ) on values of the Euler-phi function. He proves everything (definitively in some sense) **except** the classical (Carmichael) conjecture on whether this is any k for which phi(n)=k for exactly one n. In this sense, it’s a downer, that Green-Tao can similarly say “everything” about primes in linear equations, except those that were classically of most interest.

Additionally, one reason why Ford’s work in “just an Annals paper” and Green is being mentioned for the Fields Medal is that the latter has much more influence on other fields. If they had manage to prove (or phrase) their results on primes w/o the ergodic cross-over, the interest might be lower. This is also why I say that the “prime” angle is a bit stretched in some sense, though I admit that it is a meta-mathematical construct. An analogous example here might be Friedlander and Iwaniec on X^2+Y^4 being prime infinitely often — they mention that one can see (for instance) metaplectic forms bubbling in the background at various points, though they chose to write things closer in terms to “classical” analytic number theory, rather than pursue such vistas. (This seems to be their decision to try to “keep it simple” — they could likely handle all bilinear forms F(a,b^2) via similar methods (and probably a pursuant increase in notational logjam), but chose not to do so).

I guess it shows my lack of knowledge that I did not know that the general Inverse conjecture was (announced to be) proven. The Fields committee have typically (I think) only relied on published work (see my comment about Bhargava — his work on Selmer groups is joint also I think, not to mention that he still has the “lack of exceptional Lie algebras” problem and so only reaches p=5).

If Lovasz has sway, Green could well win, but I still think it is a longer shot than the others.

24. Lambert says:

Hmm to quote: I don’t think it’s fair to say that Green and Tao miss “all the interesting cases”

Whereas I said: “as they miss the interesting cases such as twin primes”

Perhaps I should say “the most interesting cases”, yet as I say above, it’s a “moving goalposts” problem — we can prove X, which is swell and nifty in and of itself, even though the classical concern (or motivating question) was to prove the related statement Y.

For instance, if someone had proven something “close” to the Poincare conjecture (pre Perelman), but not strictly as-is, then it would (or should, I say) devolve to a question of the merits strictly on the actual work, with the putative Poincare angle in the background. This said, another facet of the Green-Tao output is to re-position the relevant questions, to some extent. I would still find it more spectacular (to me personally) to prove twin primes, via whatever method (including bog-standard sieves, with little impact elsewhere).

25. Lambert says:

Finally, not to post too much, but the peculiarity that Green-Tao first did their work conjecturally (or at least the application to primes, which you seem to indicate would be a major Fields component of it), and then proved the conjectures, makes it a bit less clean, again in some meta-mathematical sense. In fact, I had previously not looked at much of their linear equations in primes particularly because: it didn’t cover twin primes, and was conjectural (involving two sets of conjectures, no less). So my feeling as to what is ground-breaking and what is technical might be flawed or biased (for instance, my impression is that there is a reasonable amount of new work in making Leibman quantitative: back in 2000 I recall, with respect to equidistribution of Heegner points perhaps though I forget, that the general idea of making Ratner’s work quantitative was generally desirable, but maybe some result of Borel was the sticking point to doing this mechanically). Maybe they need a whole “revision” project to write things the “correct” way from their current perspective (this is sort of tongue-in-cheek, though my guess is that someone will do this in a book sooner or later)?

Also, already Green-Tao use the adjective “important” for twin-primes and Goldbach in their abstract, and I really do feel they are missing the “smoking gun” in its elision.

26. Chriss says:

My guess: Ben Green, Artur Avila, Kathrin Bringmann and someone surprising! Any bets against me?

27. antonio says:

my candidates are: ben green,chistopher hacon,cedric villani
chern medal,maybe simon donaldson

28. Dear Lambert,

First, I’m sorry for the misquote.

Second, although the first Green–Tao paper on Hardy–Littlewood questions set up a conjectural framework, they did prove their conjectures in the s = 2 case (if I remember the terminology correctly)
at the same time that that paper appeared, thus giving the correct asymptotics for 4 primes in arithmetic progression;so it wasn’t *all* conjectures.

Third, the announcement of the general case of the Inverse Conjecture is made in the paper on the U^4 case.

Finally, I appreciate your thoughtful discussion of and reflections on this work.

Here are some final comments to further explain my own reaction and evaluation (“spectacular”). First, after the initial Green–Tao result on primes in A.P., I was curious about what asymptotics their result gave,
and how close they were to the truth. (Not very close, as it turned out.) Thus I was also aware of the fact that their results didn’t use much number theory, but were primarily combinatorial.

So when they came out with their first Hardy–Littlewood paper which gave a program for deriving the correct asymptotics, proved it in the length 4 case, and furthermore integrated their earlier combinatorial view-point with what seems to me to be substantial analytic number theory, I was really impressed! Being an algebraic number theorist myself, I also admired the structural framework they created, both for its apparent power and also because of its aesthetic appeal.

Not too long after the paper appeared I also had a conversation about it with Sarnak, who explained to me how to solve certain problems that he was interested in using their results. This solidified my sense of the importance of what they had done.

Of course, proving twin primes by bog-standard sieves would be really great too (!).

29. Lambert says:

I might point out a few more things.

Soon after the first Green-Tao result, it was “translated”(see http://magma.maths.usyd.edu.au/~watkins/papers/gt.pdf) into a somewhat crude re-writing in a language that avoided ergodic theory (I think a few analytic number theorists cribbed from this or something similar, at least on their first trek into the brave new additive combinatorics world). I don’t know if one could prove the later results in the same manner. I think this (rather dense) 20-page note is meant to read in conjunction with GT, as it omits almost all the motivation.

I think all the difficulties are already in the following statement: given k, are there infinitely many m,n with m+n, m+2n, m+3n, … m+kn all prime, and if so, what is the asymptotic count? The original GT proves the first part.. The later GT(Z) work(s) is to establish an asymptotic herein. The additional language with complexity of linear forms, etc., gains a bit of generality, but from the minimalist view, I don’t think is obligatory. So the challenge is: can the above “simplified statement” be proven without fancy ergodic manifold language?

On to some comments about the GT(Z) project, at all kinds of different levels. I happened to look at the Linear Equations in Primes paper more thoroughly yesterday. In some sense (to my eye), the major content for s=2 is almost totally in the companion papers, while LEP rather handles a lot of technical reductions (largely what they call a transference principle — this can be interesting by itself, but that’s a taste statement), and gives a manifesto (page 10 of the ArXiv pre-print): “We fully expect GI(s) and MN(s) to be settled shortly…” — I guess they really believed in themselves. Having a Fields medalist on board probably aided the Annals editor in assuring the acceptance this paper (as I say, I find the two s=2 papers to have the major content, and I would have needed greater elucidation of why they thought there program would work — though in retrospect I can perhaps see it a bit more). It would be interesting to know which they thought would be easier. After a bit thought, I now think that MN(s) is vaguely like showing no zeros on the 1-line (though a vast generalisation), and I guess I don’t see any reason to think that the s=2 argument would not generalize, given the concurrent work on nilmanifolds. See below for my thoughts on GI(s). I should say, though, that they really do irk me, when just when I think they are going to prove something about the “rough” piece of the Lambda-splitting in Section 12, they invoke MN(s)… (I was also the opposite to you: I cared much more about the qualitative primes in AP than the quantitative)

I am not sure I agree with the moniker “Generalized Hardy-Littlewood” though. This is probably where they first entered my doghouse. They give some hand-wavey idea that HL *could* have conjectured such a thing, had HL thought to do so — I take the opposite view, and say that HL were more interested in the “hard” 1-variable case, and basically the HL label on the multi-variable case is just borrowing the “randomness of primes” philosophy from them. I guess a pedant could even point out that the intersection of what they prove with the strict HL statement is, I think, merely the prime number theorem (with Dirichlet for AP if you include their “circle of ideas” they mention from Dickson).

I also don’t like the fact that they constantly shunt stuff to appendices, as they have many (many) arguments that are “trivial” adaptations of things in literature (sometimes adapting their own work — for instance, the GI(s) conjecture is now for polynomial sequences rather than for just linear sequences as it was first stated), and often they label their proofs as sketches when they do this. Some of this could be that I sometimes find them to sketch the things I don’t understand, and belabor those that I do. On a more personal level, they overly frequently use a construct like: “We want to show X, and we reduce this to showing Y via this 2-line argument”, and this is not my style vis-a-vis lemmas and top-down thinking. I fully understand that Tao loves to write this way, and for pedagogical purposes of understanding it might be well and good (at least for some), yet for the “scientist” who wants first and foremost to verify the result (actually, are there any such mathematicians today?), it can be a bit harrowing to sort it all out. I could almost ask for Leitfaden of implications as an aid. :) As I say, it could use a “post-GT(Z)” re-write by someone, though the principals have probably reached the burn-out stage.

As for GI(s), they do seem to allude to its complete resolution in a forthcoming paper, though the fact that they also promise to rewrite everything in nonstandard analysis (to simplify hierarchies of complexity and epsilons), plus the need to formalize the generalized calculus of polynomial brackets, not to mention the passage from “just the Heisenberg group” to more complicated Lie algebras (some of this being in preprints of others) — well, I plead scientific skepticism in being a bit cautious. I looked at how the GI(2) proof went, and it “would be nice” if they could do everything in terms of explicit manifolds as in the recurring Heisenberg “model example” (essentially upper triangular matrices), though they seem to back away from that in GTZ. I’m not sure whether they are looking at generality with manifolds for motivation, simplicity, necessity, or what.

Finally, as to whether the whole shebang is “really” about the primes, I could go either way. Certainly you can argue: they have a black box (almost completely operative by now) that turns a regularity hypothesis about some object into a theorem about it. They prove the primes are sufficiently regular, but mostly tinker with the box (such as the quantitative work on manifolds). The fact that use the Mobius function could also be a misnomer, as they could (presumably) rephrase the orthogonality for any such function that has similar properties. OTOH, the fact the “Linear equations in primes” is in Annals shows that this is their main motivator. The combinatorics angle is really due to Gowers, with all his averaging (hence the norms of his name), though I’ve never looked at his work too deeply.

30. Arctorius says:

Matthew Emerton said:
>When I wrote that this work is entirely about primes, I had two things in >mind: (a) it is about the precise Hardy–Littlewook asymptotics for linear >equations in primes, so in a direct sense it is entirely about preims; (b) the >(non)correlation result for the Mobius function and nil-sequences seems to >be squarely in the tradition of classical analytic number theory, and so in a >technical sense is “entirely about primes”.

First of all, I did originally have Green and Tao’s first work in mind when I said that it was not truly about the primes. Thanks to M. Emerton for his comments.

At the same time, (a) is a sort of trivial sense, in that, were this another set with vaguely nice properties inside a so-called “enveloping sieve”, we would get its asymptotics instead; there’s nothing special about major-arcs estimates for the primes that makes things work.

I partially agree with (b). At the same time, the estimates that Green and Tao had to prove here are essentially of the same type as

\sum_{n<=x} mu(n) e(P(n)) = o(x),

where P is a polynomial. It's my understanding that such estimates have been around for a very long time, and are not much harder than estimates of the form \sum_{n<=x} mu(n) e(alpha n) (which were hard for Vinogradov, but are extremely standard nowadays). The precise phrasing of the problem they had to solve is a bit more general.

The substance, then, in recent developments must lie in their joint work with Ziegler, and in joint work of Tao and Ziegler alone. I am looking forward to further developments in this front; so far, the inverse theorem for higher norms has only been announced.

Returning to the original topic – yes, if Bhargava's new work on Selmer groups gets out and can be checked on time, he should be a shoe-in. (I'm adding this exciting new development to all the terrific work he did in the past.)

31. Arctorius says:

>Soundarajan is similar to Bhargava in breadth versus depth, and quantum >unique ergodicity is not as important to physicists as was represented, >which takes out a “main-result” consideration.

Soundararajan has done a very great deal of deep work, besides having broadened after his (stunning) early work in classical analytic number theory. As for quantum unique ergodicity – my understanding is that it is important; I have never known Soundararajan to overpublicise his work.

32. Bristane says:

Ngo, Avila,Green will get the Medal!

But definetly not Lindenstrauss – Have you ever listened to one of his talks? Though its mathematically deep, it’s hard to follow him…

33. Arctorius says:

Lindenstrauss, Ngo, Smirnov, Villani got the medal.