Photo of Roger Heath-Brown

Oxford Mathematician Roger Heath-Brown has been awarded the Sylvester Medal by the Royal Society "for his many important contributions to the study of prime numbers and solutions to equations in integers". The Sylvester Medal is awarded annually by the Royal Society for an outstanding researcher in the field of mathematics. The award was created in memory of the mathematician James Joseph Sylvester who was Savilian Professor of Geometry at the University of Oxford in the 1880s.

Roger Heath-Brown is one of the foremost analytic number theorists of his generation. His important works on prime numbers and related topics include, among many others:

- "Heath-Brown's identity", an important way of decomposing the primes into multilinear pieces, used in many other works such as Zhang's work on bounded gaps between primes

- There are infinitely many primes of the form x^3 + 2y^3 (currently the sparsest natural sequence where one can find primes)

- if a is coprime to q, there is always a prime a (mod q) of size < q to the power 5.5

- at least one of 2,3,5 is a primitive root modulo infinitely many primes.

His contributions to solving equations in integers and rationals include, for instance:

- every nonsingular cubic form in 10 variables has a rational point (and 10 is best possible)

- every cubic form in 14 variables has a rational point

- development of "the determinant method"

- breakthrough quantitative results on the number of rational points up to a given height

Roger Heath-Brown was educated at Cambridge (a student of Alan Baker) and moved to Oxford in 1979. He was made FRS in 1993, and was twice a speaker at the International Congress of Mathematicians. He remained at Oxford throughout his career, first at Magdalen College and then, upon being promoted to a personal statutory professorship in 1999, at Worcester College. He retired in 2016. Among his many graduate students was James Maynard, who was awarded the Fields Medal in 2022.

You can watch an interview with Roger by Ben Green on occasion of his retirement (a loose term for a mathematician) .

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