The little lemma that could — the work of Ngô Bào Châu

Results in mathematics come in several flavours — theorems are the big important results, conjectures will be important results one day when they are proved, and lemmas are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?

It turn's out that Ngô's lemma, formulated in 1970 as part of the famous Langlands programme, wasn't so small after all. And after an enormous amount of mathematical theory came to rely on this unproven lemma, it got a promotion and became known as the Fundamental Lemma.

In the 1970's the mathematician Robert Langland had a grand vision that could bring together the seemingly unrelated fields of group theory, number theory, representation theory and algebraic geometry. Langlands work laid out a mathematical map connecting these diverse areas of mathematics which has lead to a large area of research known as the Langlands program. One of the most important tools in this work is the trace formula, an equation which allows arithmetic information to be calculated from geometric information, itself linking together the disparate concepts of the continuous (a property of things that can be divided into infinitisemal parts, which include geometric objects such as lines, surfaces and the the three-dimensional space we live in) and the discrete (describing things that come in whole indivisible parts, such as the whole numbers which are studied in number theory).

However in order for the trace formula to be applied in any useful way in the Langlands program, an apparently relatively simple condition, that two complicated sums were equal, needed to hold. Langlands and others assumed this condition was true and stated it as a lemma, and many results in the Langlands programme relied on this lemma being true. Langlands set a graduate student the exercise of proving this lemma, but when the student, and then many others failed to prove the result was true, it became known as the Fundamental Lemma. And it's unproven state became a thorn in the side of the Langlands programme.

Finally the thorn has been removed as Ngô has proved the Fundamental Lemma using some surprising methods. Unexpectedly he was able to use geometric objects, called Hitchin fibrations, solve this problem in pure mathematics. And not only did he prove this result, he provided a deeper understanding of this area of mathematics. Ngô not only revealed the the mathematical iceberg of which the tip was the Fundamental Lemma, he also provided a way to understand the whole ice field, said James Arthur, who explained Ngô's acheivements to the International Congress of Mathematicians after Ngô received his Fields Medal.

Ngô's work highlights the unexpected nature of mathematics, how even the smallest steps in a proof can turn out to be giant leaps in knowledge and understanding. And Ngô's career as an explorer in the mathematical wilderness has only just begun, and we all look forward to the new frontiers he will go on to discover.

You can read more about Ngô Bào Châu's work on the ICM website. And you can read more about group theory, number theory and algebraic geometry on Plus.