The work of Robert Langlands and Richard Taylor, taken together, provides us with an extraordinary unifying vision of mathematics. This vision begins with “Reciprocity”, the fundamental pillar of arithmetic of previous centuries, the legacy of Gauss and Hilbert. Langlands had the insight to imbed Reciprocity into a vast web of relationships previously unimagined. Langlands’ framework has shaped – and will continue to shape, unify, and advance – some of the most important research programmes in the arithmetic of our time as well as the representation theory of our time. The work of Taylor has, by a route as successful as it is illuminating, established – in the recent past – various aspects of the Langlands programme that have profound implications for the solution of important open problems in number theory.
For a prime number p form the (seemingly elementary) function that associates to an integer n the value +1 if n is a square modulo p, the value – 1 if it isn’t, and the value 0 if it is divisible by p. It was surely part of Langlands’ initial vision that such functions and their number theory might be relatively faithful guides to the vast number-theoretic structure concealed in the panoply of automorphic forms associated to general algebraic groups. Langlands, viewing automorphic forms as certain kinds of representations (usually infinite-dimensional) of algebraic groups, discovered a unification of the two subjects, number theory and representation theory, that has provided mathematics with the astounding dictionary it now is in the process of developing and applying. Namely, the Langlands Philosophy: a dictionary between number theory and representation theory which has the uncanny feature that many elementary representation-theoretic relationships become – after translation by this dictionary – profound, and otherwise unguessed, relationships in number theory, and conversely.