1.1. LMS Senior Whitehead Prize.

The Senior Whitehead Prize is awarded by the London Mathematical Society, in memory of Professor J H C Whitehead, a former President of the Society. It is awarded in odd-numbered years. The Senior Whitehead Prize for year X can only be awarded to a mathematician who is normally resident in the United Kingdom on 1 January of year X. The grounds for the award may include work in, influence on or service to mathematics, or recognition of lecturing gifts in the field of mathematics; the Senior Whitehead prize may not be awarded to any person who has previously received the De Morgan Medal, Pólya Prize, Senior Anne Bennett or the Naylor Prize.

1.2. Birch receives the Senior Whitehead Prize.

In the London Mathematical Society Newsletter of July 1993 the winner of the Senior Whitehead Prize was announced:

The Senior Whitehead Prize is awarded to B J Birch for his work in number theory, and in particular for his outstanding contributions to the arithmetic of elliptic curves.

1.3. The Senior Whitehead Prize Lecture.

Bryan Birch delivered the Senior Whitehead Prize Lecture Elliptic Curves over the Rationals at the meeting of the London Mathematical Society held at the Linnean Society, Burlington House, Piccadilly, London W1 on Friday 17 June 1994.

2.1. Clay Mathematics Institute's seven Millennium Problems.

The Clay Mathematics Institute gave seven Millennium Problems each with a $1 million dollar prize for a solution. Six remain unsolved (in 2025) with the Poincaré Conjecture being the only one solved. The six unsolved Millennium Problems are:

- Birch and Swinnerton-Dyer Conjecture.
- Hodge Conjecture.
- Navier-Stokes Equation.
- P vs NP.
- Riemann Hypothesis.
- Yang-Mills & the Mass Gap.

2.2. Birch and Swinnerton-Dyer Conjecture: Overview.

The Clay Mathematics Institute gives the following Overview of the Birch-Swinnerton-Dyer Conjecture.

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod $p$ to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles' proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

2.3. Birch and Swinnerton-Dyer Conjecture: Advanced.

Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers $x, y, z$ to algebraic equations like $x^{2} + y^{2} = z^{2}$.

Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in 1970 Yu V Matiyasevich showed that Hilbert's tenth problem is unsolvable, i.e., there is no general method for determining when such equations have a solution in whole numbers. But in special cases one can hope to say something. When the solutions are the points of an abelian variety, the Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behaviour of an associated zeta function $\zeta(s)$ near the point $s = 1$. In particular this amazing conjecture asserts that if $\zeta (1)$ is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if $\zeta (1)$ is not equal to 0, then there is only a finite number of such points.

3.1. The De Morgan Medal.

The De Morgan Medal is awarded by the London Mathematical Society in memory of Professor Augustus De Morgan, the Society's first President.  It is the Society's premier award and is awarded every third year (in years numbered by a multiple of three). The De Morgan Medal for year X can only be awarded to a mathematician who is normally resident in the United Kingdom on 1 January of year X. The only grounds for the award of the Medal are the candidate's contributions to mathematics.

3.2. De Morgan Medal awarded to Bryan Birch.

The following announcement appeared in the London Mathematical Society Newsletter of July 2007:

Professor Bryan Birch of the University of Oxford is awarded the De Morgan Medal in recognition of his influential contributions to modern number theory. His joint work with Peter Swinnerton-Dyer on elliptic curves created an exciting new area of arithmetic algebraic geometry: the Birch-Swinnerton-Dyer conjecture remains after 40 years one of the most tantalising problems in modern mathematics. Birch's work on Heegner points has led to huge advances in the arithmetic of elliptic curves.

3.3. Mathematician honoured for his $1 million dollar question.

The London Mathematical Society has awarded its foremost prize, the De Morgan Medal, to a mathematician who developed a problem which has yet to be solved.

The LMS Council announced that Professor Bryan Birch, of the University of Oxford, had been awarded the triennial prize in recognition of his influential contributions to modern number theory. This area of mathematics is used in ensuring the security of Pin numbers and communications generally.

In particular, Professor Birch worked with Professor Sir Peter Swinnerton-Dyer, of the University of Cambridge, to create a new area of arithmetic algebraic geometry. Together they formulated the Birch-Swinnerton-Dyer conjectures. Despite the best efforts of some of the greatest mathematical minds these remarkable conjectures are still open after 40 years and are amongst seven classic unsolved mathematical problems identified by the Clay Mathematics Institute in Cambridge, Massachusetts. The Institute is offering $1 million prizes for their proofs.

Professor John Toland, President of the LMS, said, "These remarkable conjectures are still open and a million dollar prize awaits anyone who comes up with a proof. For these conjectures, and for many other seminal contributions, we honour Professor Birch today." Professor Toland quoted mathematician Georg Cantor, who, in 1867 said, "In mathematics, the art of proposing a question must be held of higher value than solving it."

The Council for the LMS announced the De Morgan Medal and other prize winners at the Society Meeting held at University College London on 22 June. The awards were presented at the LMS Annual General Meeting on 23 November.

The De Morgan Medal.

The De Morgan Medal is awarded to Professor Bryan Birch of the University of Oxford in recognition of his influential contributions to modern number theory.

Bryan Birch's joint work with Peter Swinnerton-Dyer on elliptic curves created an exciting new area of arithmetic algebraic geometry; the Birch-Swinnerton-Dyer conjecture remains after 40 years one of the most exciting problems in modern mathematics. His work on Heegner points has led to huge advances in the arithmetic of elliptic curves.

4.1. The Sylvester Medal.

The Sylvester Medal is awarded annually for outstanding contributions 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.

4.2. Citation for the award to Bryan Birch.

His work has played a major role in driving the theory of elliptic curves, through the Birch-Swinnerton-Dyer conjecture and the theory of Heegner points.

4.3. Birch awarded Sylvester Medal.

The American Mathematical Society gave the following News in the Notices of the American Mathematical Society 67 (10) (2020), 1627-1628.

Bryan Birch, professor emeritus of the University of Oxford, has been awarded the Sylvester Medal of the Royal Society for work that "has played a major role in driving the theory of elliptic curves, through the Birch-Swinnerton-Dyer conjecture and the theory of Heegner points." The citation reads: "After writing his thesis under the supervision of Ian Cassels, Bryan Birch followed Harold Davenport in applying analytic methods to prove results about the zeros of rational polynomials in many variables. For instance, if such a polynomial of odd degree has enough variables, it will certainly have a rational zero.

"He is best known for his share in formulating the Birch and Swinnerton-Dyer conjecture. This relates the Mordell-Weil group of rational points of an elliptic curve, $E$, defined over the rationals to the behaviour of the $L$-function of $E$ near its critical point. (The conjecture is one of the Clay Millennium Problems.)

"Bryan's rediscovery of Heegner points has enabled others to (nearly) prove the conjecture when the Mordell-Weil rank is 0 or 1, although it has not yet been fully solved. The conjecture has encouraged major advances in the theory, which has surprisingly important applications to information security."

Birch is a Fellow of the AMS. He was elected a Fellow of the Royal Society in 1972 and is a recipient of the Senior Whitehead Prize and the De Morgan Medal of the London Mathematical Society.

The Sylvester Medal is awarded annually for an outstanding researcher in the field of mathematics and carries a cash award of £2,000 (approximately US$2,400).

4.4. Prestigious Honour for LMS Member.

The London Mathematical Society announced the award to Bryan Birch in their November 2020 Newsletter:

The Sylvester Medal has been awarded to LMS member Professor Bryan Birch (University of Oxford). As noted in the citation, Professor Birch's work has played a "major role in driving the theory of elliptic curves, through the Birch-Swinnerton-Dyer conjecture and the theory of Heegner points". He has been an LMS member since 1958 and a Fellow of the Royal Society since 1972. His support for the LMS has included editing the Proceedings of the London Mathematical Society from 2000 to 2002. He has also received several LMS honours including the Senior Whitehead Prize in 1993 and the Society's highest honour, the De Morgan Medal, in 2007.

4.4. Bryan Birch awarded the Royal Society's Sylvester Medal for 2020.

Oxford University announced the award to Bryan Birch on their "News" webpage on 4 August 2020:

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 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.