The Fundamental Lemma September 13, 2009
Posted by Ben Webster in Algebraic Geometry, fields medals.trackback
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.
Potentially relevant: Ngo’s paper.
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.
Surely this is supporting evidence for Doron Zielberger’s assertion that a good lemma is worth a thousand theorems.
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.
Someone has to be rewarded for Birkar-Cascini-Hacon-McKernan.
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.
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?
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.
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.
Also, what about Ben Green??
How about Bridgeland?