A Fields Medal for Ngô Bảo Châu

It had been widely predicted that Ngô Bảo Châu would be awarded a Fields medal this year. The reason was his proof of  the Fundamental Lemma for the case of reductive groups last year. This is considered a breakthrough step in the Langlands Program, a collection of far-reaching conjectures linking number theory and group representation theory.

Ngô Bảo Châu was born in Hanoi in 1972, making this the last time he would be eligible for the Fields Medal before passing his 40th birthday. Like many field medalists his talent for maths became apparent early. He was a double gold medalist at the International Maths Olympiads with a rare perfect score in one year. After becoming the youngest professor in Vietnam in 2005 he moved to the US and is now at the IAS in Princeton.

in 2004 Ngô Bảo Châu proved the fundamental lemma for unitary groups with Gérard Laumon, but it is his more general result from 2008 when he proved the lemma for all reductive groups that made him a mathematical legend. This is just the kind of singular achievement that the Fields medal likes to recognise.

A result in mathematics is called a lemma if it is a step required towards proving a bigger and more important theorem. Typically a lemma is easier to prove but in the case of the fundamental lemma this was far from true. In the 1960s, Robert Langlands had found a strategy for proving a set of important results known now as the Langlands conjectures, relating Galois theory from algebra to automorphic forms in analysis.  The proves were incomplete and one step needed to bridge the gaps was to show that certain identities on reductive groups hold. This was the lemma that Ngô Bảo Châu finally proved. As a consequence many important relationships of the Langlands program were then also known to hold. Mathematician Peter Samak described the feeling of the those working in the field by saying “It’s as if people were working on the far side of the river waiting for someone to throw this bridge across. And now all of sudden everyone’s work on the other side of the river has been proven.”

The Langlands program remains one of the hardest areas of mathematics to penetrate because it requires expertise in different branches from number theory to algebra, special functions and algebraic geometry. Since Langlands initial insights it has gradually become clear that the conjectures are important to understanding of many beautiful results. They are even thought by some to have deep applications in theoretical physics.

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