# The Abel Prize 2020

We give below the part of the citation for the Abel Prize 2020 which gives an overview of the work of Hillel Furstenberg and Gregory Margulis. We also give that part of the citation which refers specifically to the work of Furstenberg.

The Abel Prize 2020

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2020 to Hillel Furstenberg, Hebrew University of Jerusalem, Israel and Gregory Margulis, Yale University, New Haven, CT, USA
for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.
A central branch of probability theory is the study of random walks, such as the route taken by a tourist exploring an unknown city by flipping a coin to decide between turning left or right at every cross. Hillel Furstenberg and Gregory Margulis invented similar random walk techniques to investigate the structure of linear groups, which are for instance sets of matrices closed under inverse and product. By taking products of randomly chosen matrices, one seeks to describe how the result grows and what this growth says about the structure of the group.

Furstenberg and Margulis introduced visionary and powerful concepts, solved formidable problems and discovered surprising and fruitful connections between group theory, probability theory, number theory, combinatorics and graph theory. Their work created a school of thought which has had a deep impact on many areas of mathematics and applications.

Starting from the study of random products of matrices, in 1963, Hillel Furstenberg introduced and classified a notion of fundamental importance, now called Furstenberg boundary. Using this, he gave a Poisson type formula expressing harmonic functions on a general group in terms of their boundary values. In his works on random walks at the beginning of the '60s, some in collaboration with Harry Kesten, he also obtained an important criterion for the positivity of the largest Lyapunov exponent.

Motivated by Diophantine approximation, in 1967, Furstenberg introduced the notion of disjointness of ergodic systems, a notion akin to that of being coprime for integers. This natural notion turned out to be extremely deep and have applications to a wide range of areas including signal processing and filtering questions in electrical engineering, the geometry of fractal sets, homogeneous flows and number theory. His "×2 ×3 conjecture" is a beautifully simple example which has led to many further developments. He considered the two maps taking squares and cubes on the complex unit circle, and proved that the only closed sets invariant under both these maps are either finite or the whole circle. His conjecture states that the only invariant measures are either finite or rotationally invariant. In spite of efforts by many mathematicians, this measure classification question remains open. Classification of measures invariant by groups has blossomed into a vast field of research influencing quantum arithmetic ergodicity, translation surfaces, Margulis's version of Littlewood's conjecture and the spectacular works of Marina Ratner. Considering invariant measures in a geometric setting, Furstenberg proved in 1972 the unique ergodicity of the horocycle flow for hyperbolic surfaces, a result with many descendants.

Using ergodic theory and his multiple recurrence theorem, in 1977, Furstenberg gave a stunning new proof of Szemerédi's theorem about the existence of large arithmetic progressions in subsets of integers with positive density. In subsequent works with Yitzhak Kaztnelson, Benjamin Weiss and others, he found higher dimensional and far-reaching generalisations of Szemerédi's theorem and other applications of topological dynamics and ergodic theory to Ramsey theory and additive combinatorics. This work has influenced many later developments including the works of Ben Green, Terence Tao and Tamar Ziegler on the Hardy-Littlewood conjecture and arithmetic progressions of prime numbers.

Last Updated December 2023