Why is the Fundamental Lemma, ‘fundamental’?
A large class of mathematical results rely on Chau’s award winning proof of the ‘fundamental lemma’. The proof ensures that a set of essential conditions, forming the basis of other mathematicians’ work, are true. More simply, without the proof of the ‘fundamental lemma’ many parts of mathematics would remain undecidable. That is, their truth, or their falsehood, could not be determined.
A non-mathematical example for undecidability is going skydiving without checking your parachute. Imagine how important it would be to know that you have a working parachute before jumping out of a plane. The parachute in this analogy is the ‘proof of the fundamental lemma’. Many mathematicians have taken the risk and have jumped without checking their parachutes. Luckily, Chau’s work has eased everyone’s worries. Also, this mental image should be taken with a grain of salt; no known mathematicians have been that reckless, in real life.
Chau’s result is revolutionary since it strikes at the foundations of mathematics. It provides confidence that many mathematicians are working in the right direction and ensures that their mathematical machinery is working properly. The methods of his proof were mainly algebro-geometric, but the consequences have a significant impact on all mathematical fields, from analytic number theory to mathematical physics. One reason is that the methods are transferable and can be used to produce other results. It may even provide the proper technique for many questions that have not even been asked yet. Another reason is that it bridges the gap between fields and essentially serves as a translator back and forth between the languages of number theory and harmonic analysis. [See Edward Frenkel’s Interview]
Another post will deal with the fact that results similar to Chau’s have been posed many times in the past and I’ll explore questions have been asked systematically in the Langlands Program. As mentioned in a previous post, a half-dozen Fields Medalists have been involved with these kinds of questions and many more have reaped the benefits of these breakthroughs.
The researchers invited to attend the inaugural Fields Medal Symposium have made significant contributions to Ngo Bao Chau’s areas of research. In particular, Chau’s research spans the mathematical areas of algebraic geometry, group theory, and automorphic representations.
Another very good treatment of Chau’s result is given in the ICM2010 Archives
A technical summary, The laudations by James Arthur
A non-technical summary, The work profile by Julie Rehmeyer