Fields Medal 2010

Ngo Bau Chau was awarded the Fields medal on the 19th August 2010 at the International Mathematics Congress (ICM) at Hyderabad, for his research on the Langlands programme and the proof of the fundamental lemma. (Consult the ICM2010 website.)

What is the fundamental Lemma?

From the beginning of his tenure as Chargé de Recherche at the LAGA, University Paris 13, Ngo Bau Chau's ambitious aim was to prove what the specialists call The Fundamental Lemma. This asserts the existence of an equality between two explicit integrals, which can only be evaluated in special cases. This is one of the corner stones of the Functoriality, known as the Langlands programme, which establishes links between the branches arithmetic, analysis and geometry of pure mathematics.

In the simplest case, as for the famous Delta function of Ramanujan, a modular form is an analytic function on the Poincaré half plane (the complex numbers with positive imaginary part) which is preserved under the action of a group of Poincaré transformations and which satisfies a growth condition. More generally, a modular form can be considered as a function on the moduli space of isomorphism classes of elliptic curves. Automorphic forms are generalizations of the notion of modular forms to the case of several complex variables. Automorphic forms are the fundamental objects of Langland's programme associated to a Lie group; from this viewpoint, modular forms are automorphic forms associated to a linear group of dimension two.

One can associate L-functions to automorphic forms; these are meromorphic complex functions which satisfy functional equations, generalizing the Riemann zeta function. The heart of the Langlands programme is to associate L-functions to different mathematical objects: representations (of Galois groups and of Lie groups), diophantine equations (a diophantine equation is an algebraic equation with integral coefficients of which the integral solutions are studied) and to relate these objects via their L-functions.

A formula developed by James Arthur at the end of the last century, the trace formula, is the fundamental tool for establishing relations between these L-functions. The fundamental lemma, discovered in 1979 by Labesse and Langlands and then formulated in full generality by Langlands and Shelstad in 1987, englobes a whole series of equalities between orbital integrals associated to different groups. In certain cases, the fundamental lemma can be proved by combinatorial methods, but it seems likely that the general result is out of reach of such techniques.

At the beginning of the 2000s, Goresky, Kottwitz et MacPherson proposed a novel approach; the principle was to interpret the different integrals using the points defined over finite fields of certain geometric objects. Using this approach, they succeeded in proving the fundamental lemma, modulo a 'purity' conjecture concerning a certain algebraic invariant of the cohomology of these objects, which they proved in a special case.

Many mathematicians attempted unsuccessfully to prove this purity conjecture. The essential contribution of Ngo Bau Chau, achieved during his years at Paris 13, was to introduce new geometrical objects, studied by Nigel Hitchin in theoretical physics. These are a deformation of the Goresky, Kottwitz, MacPherson geometry, namely a family of geometrical objects containing that of interest and also a simpler object, for which the purity conjecture is known. The proof relies on transporting information within this family, programme which was completed by Ngo Bau Chau in collaboration with Gérard Laumon.