# The 2007 Prize in Mathematical Sciences

## Robert LanglandsRichard Taylor

for initiating and developing a grand unifying vision of mathematics that connects prime numbers with symmetry.

## The Contribution

The Shaw Prize in Mathematical Sciences 2007 will be awarded in equal shares to Robert Langlands and Richard Taylor.

Robert Langlands initiated a unifying vision of mathematics that has greatly extended the legacy of the mathematics of previous centuries, connecting prime numbers with symmetry. This unification, which grew out of the Reciprocity Theory of Gauss and Hilbert, is now referred to as the Langlands program. It provides a direction of research which has guided mathematicians over the past forty years and will continue to do so for years to come.

## An Essay on the Prize

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.

Robert Langlands

Robert Langlands, born 1936 is currently a Professor at the Institute for Advanced Study (IAS) at Princeton. Born in Canada, Professor Langlands attended the University of British Columbia gaining an undergraduate degree in 1957 and the MSc in 1958. In 1960 he received his PhD from Yale University. He taught at Princeton and Yale before moving to IAS in 1972. He is a Fellow of the Royal Society of London, the Royal Society of Canada and the US National Academy of Sciences.

12 June 2007, Hong Kong

Autobiography

Richard Taylor

Richard Taylor, born 1962 is currently the Herchel Smith Professor of Mathematics at Harvard University, a post he has held since 2002. Professor Taylor was born in England. He received his BA from Cambridge University in 1984 and his PhD from Princeton University 4 years later. He taught at Cambridge University from 1989 to 1995 and held the Savilian Chair of Geometry at Oxford University from 1995 to 1996. He is a Fellow of the Royal Society of London.

12 June 2007, Hong Kong

Autobiography