01 · A life in mathematics
The person behind the work
Ngô Bảo Châu was born in Hanoi and trained as a mathematician in France. At the 2010 medal he was affiliated with Université Paris-Sud and was a long-term member at the Institute for Advanced Study; his Chicago professorship began the following month. He proved the fundamental lemma in the Langlands program, a difficult identity comparing orbital integrals. His proof transformed the problem through algebraic geometry, particularly the Hitchin fibration and the geometry of its fibers. The result removed a central obstacle to comparing trace formulas, giving firm foundations to many previously conditional links between automorphic forms and arithmetic.
The contribution, explained
The Langlands program seeks deep correspondences among number theory, group representations and automorphic forms. Comparing trace formulas is a powerful way to build such correspondences, but it requires exact local identities between orbital integrals. Despite its modest name, the fundamental lemma resisted proof for decades. Ngô transformed the comparison into algebraic geometry using the Hitchin fibration, a family whose fibers encode relevant information. Studying the geometry and cohomology across the family made a previously opaque numerical equality accessible. The proof established the needed identity under its precise formulation and removed a major obstruction to the machinery of endoscopy and trace-formula comparison.
Why it matters
The theorem secured foundations for many results that had depended on the fundamental lemma and enabled further progress in the Langlands program. Its geometric methods also created new connections among representation theory, arithmetic and the study of moduli spaces.