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The Shaw Prize in Mathematical Sciences 2024 is awarded to
Peter Sarnak,
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.

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.
https://www.shawprize.org/laureates/2024-mathematical-sciences/?type=Contribution

The Shaw Prize in Mathematical Sciences 2024 is awarded to
Peter Sarnak,
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.

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.
https://www.shawprize.org/laureates/2024-mathematical-sciences/?type=Contribution