The Shaw Prize in Mathematical Sciences 2024 is awarded to Peter Sarnak, Gopal Prasad Professor, School of Mathematics, Institute for Advanced Study and Eugene Higgins Professor of Mathematics, Princeton University, USA, for his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, analysis, combinatorics, dynamics, geometry and spectral theory.

A natural number is called a prime number if it is larger than 1 and is not the product of two strictly smaller natural numbers which themselves are larger than 1. For example, 2 is a prime number, but 4 = 2 × 2 is not. Euclid’s theorem (circa 300 BCE) asserts that any natural number other than 0 and 1 is the product of prime numbers, and that there are infinitely many prime numbers. The study of the distribution of the prime numbers is a core topic in Number Theory.

The search for prime numbers has been a central theme in number theory since the ancient Greeks. One looks for polynomial functions f(x) such that f(x) is prime for infinitely many integers x. Euclid’s theorem says that f(x) = x is one such function. One may enlarge the problem by requiring that f(x) be almost prime valued, that is, the product of a bounded number of primes for infinitely many integers x. For example, the Twin Prime Conjecture is equivalent to the statement that f(x) = x(x+2) is a product of two primes for infinitely many integers x. The Chinese mathematician Jingrun Chen (1973), using Brun’s combinatorial sieve, showed that this function has at most 3 prime factors for infinitely many integers x. One may also restrict the set of x considered by requiring them to lie in a sparse subset of the integers. A similar problem can be posed for any polynomial with integer coefficients in several variables.

Sarnak pioneered the search for almost prime values of polynomials in sparse subsets arising as the orbit of a thin group. A thin group is a subgroup of an arithmetic group with a Goldilocks property: it is neither too large (being of infinite index) nor too small (having the same Zariski closure as the arithmetic group). Thin groups arise very naturally in pure and applied mathematics. For example, the symmetry group of integral Apollonian circle packings is a thin group. In addition, there is an abundance of Kleinian groups, or more generally monodromy groups of differential equations, that are thin groups.

Expanders are highly connected sparse graphs widely used in computer science. Foreseeing how the expander property of finite quotients of a thin group could be used to produce almost primes, Sarnak developed the affine sieve. Sarnak, together with Bourgain and Gamburd, produced expanders out of some thin groups. The construction relies on earlier foundational work by Sarnak and Xue in which they showed a relation between the minimal dimension of representations of finite linear groups and expanders.

Sarnak, together with Bourgain and Gamburd, obtained a precise counting and equidistribution result for integral vectors on an orbit of a thin group which take almost prime values when one applies a given polynomial function to them.

Sarnak, together with Golsefidy, established that, under some natural hypotheses, an integral polynomial function produces almost primes in a Zariski dense subset of a thin orbit.

Sarnak’s introduction of combinatorial and ergodic theoretical methods to Diophantine problems has had a profound impact. His original and deep vision has launched a vast research programme that brings together number theory, combinatorics, analysis, dynamics, geometry and spectral theory.

Mathematical Sciences Selection Committee
The Shaw Prize

21 May 2024, Hong Kong