A grand unified theory of mathematics which includes the search for a generalization of Artin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions of number 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 over finite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of matrices (Mackenzie 2000).
In 1998, three mathematicians proved Langlands' conjectures for local fields, and in a November 1999 lecture at the Institute for Advanced Study at Princeton University, L. Lafforgue presented a proof of the conjectures for function fields. This leaves only the case of number fields as unresolved (Mackenzie 2000).
Langlands was a co-recipient of the 1996 Wolf Prize for the web of conjectures underlying this program, and Lafforgue shared the 2002 Fields Medal for his progress on Langlands' program.
