The Shaw Prize in Mathematical Sciences for 2007 is shared between Robert Langlands of the Institute for Advanced Study in Princeton and Richard Taylor of Harvard University.

For more than 2000 years mathematicians have studied numbers and have shown that all whole numbers can be factorized into products of prime numbers, numbers such as 2, 3, 5, 7 …. which cannot be factorized further. Thus the primes play the role of the elements in the periodic table on display in chemical laboratories. Unlike the elements however the primes never stop. There are infinitely many of them. Nor do they have the simple periodic pattern of the chemical elements, their behaviour is more subtle. For this reason, although prime numbers are discrete quantities, their collective properties are best studied by the continuous methods of analysis, as shown in the 19th century by the great Bernhard Riemann.

The theory of numbers, which is to a large extent the study of the primes, lies at the heart of mathematics. The difficulty of its problems have led mathematicians to devise a whole range of ideas and techniques which have then been taken up and proved fruitful in other areas.

While number theory has been studied purely for its own sake, with no practical applications in view, it now plays a crucial part in modern cryptology, the encoding of information for security purposes, particularly in the financial field. Your bank account is, in a sense, protected by prime numbers.

Another key theme in mathematics is the study of symmetry which was inherent in the work of Euclid and Plato on regular solids, such as the cube or the tetrahedron, but took on a more abstract algebraic form in the early 19th century with the work of Galois on solutions of algebraic equations. These abstract symmetries involving the interchange of symbols (as with the two choices of a square root) can be represented geometrically in space of 2, 3 or more dimensions, giving rise to the theory of representations. Over the past 50 years it has become clear that symmetry, through its representations, is the fundamental principle that explains the physical structure of matter.

Robert Langlands, a Canadian from British Columbia, currently at the Institute for Advanced Study in Princeton, has over the past 40 years developed a grand scheme which relates prime numbers to symmetry, based on the theory of representations. This Langlands programme subsumes the great work in number theory of past giants such as Gauss, Riemann and Hilbert, and provides a unifying vision of large parts of mathematics. It lays out a direction of work for mathematicians for many years to come.

Richard Taylor, originally from Cambridge, England, and now at Harvard University is a leading figure of the younger generation who has shown how to carry forward significant parts of the Langlands programme, using a whole variety of methods. He was a junior partner in the celebrated solution by Andrew Wiles of Fermat's Last Theorem (recognized in the Shaw Prize of 2005), and he has collaborated with many colleagues to obtain important new results in number theory. Among these is his extraordinary work on the famous 40-year old Sato-Tate conjecture.

The work of Robert Langlands and Richard Taylor demonstrates the profundity and rigour of modern number theory. Together they amply deserve the honour of the Shaw Prize.