The Fundamental Lemma September 13, 2009Posted by Ben Webster in Algebraic Geometry, fields medals.
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.