数学苑

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2008, worth NOK 6,000,000 (close to US$ 1.2. mill, EURO 750,000) to

John Griggs Thompson Graduate Research Professor, University of Florida

and

Jacques Tits Professor Emeritus, Collège de France

"for their profound achievements in algebra and in particular for shaping modern group theory"


Modern algebra grew out of two ancient traditions in mathematics, the art of solving equations, and the use of symmetry as for example in the patterns of the tiles of the Alhambra. The two came together in late eighteenth century, when it was frst conceived that the key to understanding even the simplest equations lies in the symmetries of their solutions. This vision was brilliantly realised by two young mathematicians, Niels Henrik Abel and Evariste Galois, in the early nineteenth century. Eventually it led to the notion of a group, the most powerful way to capture the idea of symmetry. In the twentieth century, the group theoretical approach was a crucial ingredient in the development of modern physics, from the understanding of crystalline symmetries to the formulation of models for fundamental particles and forces.

In mathematics, the idea of a group proved enormously fertile. Groups have striking properties that unite many phenomena in different areas. The most important groups are fnite groups, arising for example in the study of permutations, and linear groups, which are made up of symmetries that preserve an underlying geometry. The work of the two laureates has been complementary: John Thompson concentrated on fnite groups, while Jacques Tits worked predominantly with linear groups.

Thompson revolutionised the theory of fnite groups by proving extraordinarily deep theorems that laid the foundation for the complete classifcation of fnite simple groups, one of the greatest achievements of twentieth century mathematics. Simple groups are atoms from which all fnite groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an
even number of elements. Later Thompson extended this result to establish a classifcation of an important kind of fnite simple group called an N-group. At this point, the classifcation project came within reach and was carried to completion by others. Its almost incredible conclusion is that all fnite simple groups belong to certain standard families, except for 26 sporadic groups. Thompson and his
students played a major role in understanding the fascinating properties of these sporadic groups, including the largest, the so-called Monster.

Tits created a new and highly infuential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classifcation of algebraic and Lie groups as well as fnite simple groups, to Kac-
Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits’s geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated “Tits alternative”: every fnitely generated linear group is either virtually
solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications.

The achievements of John Thompson and of Jacques Tits are of extraordinary depth and infuence. They complement each other and together form the backbone of modern group theory.