Langlands Program
Robert Langlands's letter to André Weil (January 1967) conjectures a deep link between number theory and the harmonic analysis of symmetry groups, a map that still guides the field.
Open in the interactive tree →Langlands proposed that Galois representations, which encode how the solutions of equations are permuted, correspond to automorphic forms, Fourier-like functions on groups; his functoriality conjecture relates such forms across different groups. The Langlands-Tunnell theorem of the early 1980s was a step in Wiles's proof of Fermat's Last Theorem. The correspondence is proved for function fields (Laurent Lafforgue, 1998) and in a geometric version (Gaitsgory and collaborators, 2024), but the original number-field conjectures are open. Langlands received the Abel Prize in 2018.
Prerequisites
- Number Theory & Group Theory1801Galois groups, the symmetries of equations, are one side of the correspondence.
- Fourier Analysis1822Automorphic forms generalise Fourier analysis to groups.
- Abstract Algebra (Noether)1921-1931Representation theory of groups supplies its language.
Unlocks
- Fermat's Last Theorem Proved1995Wiles used the Langlands-Tunnell theorem, a proven case of the program.
- Geometric Langlands Proved2024The geometric version of the conjectures first posed in Langlands's 1967 program.