Ngô Bảo Châu

Ngô Bảo Châu is a Vietnamese-French mathematician whose proof of the fundamental lemma advanced the Langlands program. He received the 2010 Fields Medal.

Fundamental lemma: A family of identities relating orbital integrals in harmonic analysis on reductive groups. His proof established this conjectured identity, a central obstacle in the Langlands program.
Hitchin fibration: A map from the moduli space of Higgs bundles to a space of characteristic-polynomial data. Its geometry provided the setting for the argument that established the fundamental lemma.
Robert Langlands: A Canadian-American mathematician who proposed a broad network of conjectures linking number theory and representation theory. His program provided the framework in which the fundamental lemma became a central conjecture.
Arthur–Selberg trace formula: An identity relating spectral data of automorphic forms to geometric data from conjugacy classes. The fundamental lemma permits comparisons of trace formulas across related groups.
Langlands program: A web of conjectures connecting number theory, representation theory, and harmonic analysis. The fundamental lemma supplies a key comparison needed by many parts of this program.
Higgs bundle: A vector bundle equipped with a field-valued differential form satisfying an integrability condition. Moduli spaces of Higgs bundles supply geometric structures used in the proof.
James Arthur: A Canadian mathematician known for developing the trace formula and the Arthur–Selberg theory. His work clarified the fundamental lemma’s role in the trace formula and its applications.
Endoscopic classification: The organization of automorphic representations using transfers from endoscopic groups. The lemma removes a major obstacle to establishing these transfers and classifications.
Orbital integral: An integral formed by averaging a function over a conjugacy class in a group. Orbital integrals are the quantities compared by the fundamental lemma.
Algebraic geometry: The study of geometric objects defined by polynomial equations. Geometric methods made the fundamental lemma accessible through moduli spaces and their fibers.

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