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Langlands Program
A grand unified theory of mathematics which includes the search for a generalization ofArtin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions ofnumber fields. In a January 1967 letter to André Weil, Langlands proposed that the mathematics of algebra (Galois representations) and analysis (automorphic forms) are intimately related, and that congruences overfinite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means ofmatrices (Mackenzie 2000).
In 1998, three mathematicians proved Langlands' conjectures forlocal fields, and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures forfunction fields. This leaves only the case ofnumber fields as unresolved (Mackenzie 2000).
Langlands was a co-recipient of the 1996Wolf Prize for the web of conjectures underlying this program, and Lafforgue shared the 2002Fields Medal for his progress on Langlands' program.
See also
Artin's Reciprocity Theorem,Langlands Reciprocity,Reciprocity Theorem,Taniyama-Shimura ConjectureExplore with Wolfram|Alpha
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References
American Mathematical Society. "Langlands and Wiles Share Wolf Prize."Not. Amer. Math. Soc.43, 221-222, 1996.Knapp, A. W. "Group Representations and Harmonic Analysis from Euler to Langlands."Not. Amer. Math. Soc.43, 410-415, 1996.Mackenzie, D. "Fermat's Last Theorem's Cousin."Science287, 792-793, 2000.Referenced on Wolfram|Alpha
Langlands ProgramCite this as:
Weisstein, Eric W. "Langlands Program."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/LanglandsProgram.html