Langlands conjectures

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.

Artin's reciprocity theorem | Langlands reciprocity | reciprocity theorem | Taniyama-Shimura conjecture