Oxford Mathematician Bryan Birch has been awarded the Royal Society's Sylvester Medal for 2020 for his work in driving the theory of elliptic curves through the Birch--Swinnerton-Dyer conjecture and the theory of Heegner points. The Birch--Swinnerton-Dyer conjecture is one of the Clay Mathematics Institute Millennium Problems.

The Sylvester Medal is awarded annually for an outstanding researcher in the field of mathematics. The award was created in memory of the mathematician James Joseph Sylvester FRS who was Savilian Professor of Geometry at the University of Oxford in the 1880s. It was first awarded in 1901. The medal is of bronze, is now awarded annually and is accompanied by a gift of £2,000.

Bryan Birch was educated at Trinity College, Cambridge where as a doctoral student he proved Birch's theorem, one of the results to come out of the Hardy–Littlewood circle method; it shows that odd-degree rational forms in a large enough set of variables must have zeroes.

He then worked with Peter Swinnerton-Dyer on computations relating to the Hasse–Weil L-functions of elliptic curves. They formulated their conjecture relating the rank of an elliptic curve to the order of a certain zero of an L-function; it has been an influence on the development of number theory since the mid 1960s. They later introduced modular symbols.

In later work he contributed to algebraic K-theory (Birch–Tate conjecture). He then formulated ideas on the role of Heegner points (he had been one of those reconsidering Kurt Heegner's original work, on the class number one problem, which had not initially gained acceptance). Birch put together the context in which the Gross–Zagier theorem was proved. He was elected a Fellow of the Royal Society in 1972; was awarded the Senior Whitehead Prize in 1993 and the De Morgan Medal in 2007. In 2012 he became a fellow of the American Mathematical Society.