The  Shaw  Prize  in  Mathematical  Sciences  for 2014  is  awarded  to George Luzstig, Abdun-Nur Professor of Mathematics at the Massachusetts Institute of Technology, for his fundamental contributions to algebra, algebraic geometry, and representation theory, and for weaving these subjects together to solve old problems and reveal beautiful new connections.

For more than two hundred years, symmetry groups have been at the centre of mathematics and its applications: in Fourier’s work on the heat equation in the early 1800s; in Weyl’s work on quantum mechanics in the early 1900s; and in the approach to number theory created by Artin and Chevalley. These classical works show that answers to almost any question involving a symmetry group lie in understanding its realizations as a group of matrices; that is in terms of its representations.

Starting with his early work in the 1970s and 1980s, in part jointly with Deligne, Lusztig gave a complete description of the representations of finite Chevalley groups, these being the building blocks of finite symmetry groups. The Deligne–Lusztig description uses the topology and geometry of Schubert varieties. The latter were introduced in the nineteenth century as a tool to count solutions of algebraic equations.

The vision of this work is that the algebraic subtleties of representation theory correspond perfectly to the geometric/topological subtleties of Schubert varieties. This vision has grown into a broad and powerful theme in Lusztig’s work: he has shown that many central problems in representation theory — including those of real and p-adic Lie groups, which are the language of applications from number theory to mathematical physics — can be related to topology and geometry by means of Schubert varieties. This idea is at the heart of many exciting recent developments, for example in progress toward the Langlands programme in automorphic forms.

Representations are complicated, as are the Schubert varieties to which they are related. Beginning in a 1979 paper with David Kazhdan, and continuing through his most recent work, Lusztig has found combinatorial tools to describe their topology and geometry. (These tools are easy to describe, but had not been used previously in mathematics.) His ideas have guided and inspired the development of perverse sheaves, a tool for studying the topology of general singular algebraic varieties.

These tools, in the hands of Lusztig and of hundreds of other mathematicians, have made possible a depth of understanding of representations and of Schubert varieties that was unimaginable before his work.

Mathematical Sciences Selection Committee
The Shaw Prize
27 May 2014   Hong Kong