Robert Langlands

Robert Langlands is a Canadian-American mathematician who proposed the Langlands program, a network of conjectures connecting number theory, harmonic analysis, and representation theory.

Number theory: The study of integers, primes, and the structures and patterns governing them. Langlands’s conjectures recast arithmetic questions in terms of symmetries and representations.
Langlands correspondence: A proposed correspondence between Galois representations and automorphic representations. This is the program’s central bridge between arithmetic symmetry and harmonic analysis.
André Weil: A French mathematician whose work shaped modern number theory, algebraic geometry, and representation theory. Weil’s ideas and correspondence with Langlands helped frame the program’s early questions.
Modularity theorem: The theorem that every elliptic curve over the rational numbers corresponds to a modular form. Its proof established a striking instance of the kind of correspondence Langlands envisioned.
James Arthur: A Canadian mathematician known for the trace formula and contributions to the Langlands program. Arthur’s trace formula work advances the study of automorphic representations and functoriality.
Representation theory: The study of mathematical structures through their actions on vector spaces. It supplies the language for translating arithmetic symmetries into linear algebra.
Functoriality: A conjectural transfer of automorphic representations along homomorphisms between reductive groups. It extends the program by predicting how representations move between groups.
Harish-Chandra: An Indian-American mathematician who developed foundational representation theory for semisimple Lie groups. Langlands extended methods from Harish-Chandra’s theory of representations.
Fermat's Last Theorem: The theorem that no positive integers satisfy xⁿ + yⁿ = zⁿ for n greater than two. Wiles proved it using modularity results arising from the arithmetic of elliptic curves.
Robert P. Langlands: A Canadian-American mathematician who developed the Langlands program and made contributions to automorphic forms. This closely named mathematician is distinct from Robert Langlands and is not a separate person.

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Langlands programRelated: His 1967 letter to André Weil set out the program’s foundational ideas.