John G. Thompson

John G. Thompson Winner of Wolf Prize in Mathematics - 1992
John G. Thompson

THE 1992 WOLF FOUNDATION PRIZE IN MATHEMATICS


The Mathematics Prize Committee has unanimously selected the following two scientists to equally share the Wolf Prize for 1992:


John G. Thompson
University of Cambridge
Cambridge, U.K.


for his profound contributions to all aspects of finite group theory and connections with other branches of mathematics.

Lennart Carleson
University of Uppsala
Uppsala, Sweden, and
University of California,
Los Angeles, California, USA

for his fundamental contributions to Fourier analysis, complex analysis, quasiconformal mappings and dynamical systems;

Professor Lennart Carleson fundamental contributions to Fourier analysis, complex analysis, quasiconformal mappings, and dynamical systems have clearly established his position as one of the greatest analysts of the twentieth century. His 1952 Acta paper on sets of uniqueness for various classes of functions was the breakthrough paper in that area. The 1958 and 1962 papers on interpolation and the corona problem not only solved the Corona Conjecture but introduced a host of new methods and concepts (e.g. Carleson measures, the corona construction, and the relations to interpolation). These concepts are now central to modern Fourier analysis as well as complex analysis in both one and several variables.

Carleson´s celebrated solution of the Lusin conjecture in 1965 gave a dazzling display of his technical mastery and proved the now famous result that the Fourier series of an L² function on [0,1] converges almost everywhere. In 1972 Carleson proved that in dimension two Bochner-Riesz means of any order are LP bounded, 4/3 ≤ p ≤ 4. The methods introduced are again of fundamental importance to this area of Fourier analysis.

In 1974 he proved that a quasiconformal selfmap of R³ can be extended to be quasiconformal in R4. The earlier known cases of R and R² can be solved by elementary arguments; the deep methods he introduced have now been modified so as to work in arbitrary dimension.

In 1984 Carleson and Benedicks introduced a new method to study chaotic behavior of
the map 1 → ax², and in 1988 they extended this method in a tour de force to prove that the Henon map (x ,y) → (1+y –ax, bx) exhibits 'strange attractors” for a nonempty (even positive measure) set of parameter values. This historic paper has opened an entire area in dynamical systems.

Professor John G. Thompson’s work has changed the face of finite group theory. Already in his thesis he solved a long-standing conjecture reaching back to work of Frobenius at the turn of the century: if a finite group has a fixed-point-free automorphism of finite order, then the group is nilpotent. The solution was obtained by the introduction of novel and highly original ideas. He next turned his attention to the classification of the finite simple groups. The first astonishing achievement was his joint Mark with Walter Feit in which they prove that a finite non-abelian simple group must have even order. Thompson went on to classify the finite simple groups in which every soluble subgroup has a soluble normalizer. This work is a key element in the collective effort that led to the classification of finite simple groups.

In the late 1970´s he took up McKay´s remarkable observation, that there is a connection between the Fischer-Griess group and the modular j-function to formulate a series of conjectures relating modular functions and finite sporadic simple groups. These have now been verified and have led to deep and important questions which will occupy mathematicians for some time to come.

Also during this period he significantly contributed to coding theory and the theory of finite projective planes. The recent solution of the classical problem of the non-existence of a plane of order 10 owes much to his efforts.

During the past few years he has investigated the problem of constructing Galois groups over number fields, especially Q. The starting point here is Hilbert´s irreducibility Theorem and Thompson´s work may well be the most important advance since Hilbert´s time.

The penetrating power of Thompson´s genius is astonishing; his contributions to group theory and related subjects are of enduring significance.