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. The ancient Greek mathematician Euclid proved around 300 BCE 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 French mathematician de Polignac proposed in 1849 a still unsolved problem in number theory stating that any even number can be expressed as the difference of two consecutive primes, and this can be achieved in infinite ways. If the even number is chosen to be 2, it asserts that there are infinitely many pairs (p, p+2) for which both p and p+2 are prime numbers, such as for example (5,7) or (17,19). This is called the Twin Prime Conjecture.

The Chinese mathematician Jingrun Chen made in 1973 a huge step towards the Twin Prime Conjecture. He proved that if h is a positive even integer, there are infinitely many prime numbers p such that p+h is a product of at most two prime numbers. In our days, there are several statues of Jingrun Chen in China, and a prize named after him for young mathematicians. Chen used sieve methods to count the cardinality of prime or almost prime numbers (that is a bounded product of such). These two concepts, sieve methods and almost primes, take us to the foundations of the work of Sarnak, the 2024 Shaw Laureate for mathematics.

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 enlarges the problem by requiring that f(x) be almost prime valued. 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. 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, has 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.

12 November 2024 Hong Kong