# Richard Ewen Borcherds

### Quick Info

Cape Town, South Africa

**Richard Borcherds**is a South African born mathematician known for his work in lattices, group theory and infinite-dimensional algebras, for which he was awarded the Fields Medal in 1998.

### Biography

**Richard Borcherds**'s father, Peter Howard Borcherds, had studied electrical engineering for his B.Sc. but then his interests moved towards mathematical physics. After the award of an M.Sc. and Ph.D. in physics, Peter Borcherds became a lecturer in physics at the University of Cape Town. Peter had married Margaret Elizabeth Greenfield. Peter and Margaret had four children; two of Richard's three brothers went on to become mathematics teachers, Michael Borcherds being particularly well known as the lead developer of the mathematics teaching package GeoGebra. When Richard was about a year old his parents left South Africa and went to Aldermaston in England. Peter Borcherds went on to became a lecturer in physics at Birmingham University; Peter's particular interests are the use of computers in physics and the involvement of scientists in politics.

Richard attended King Edward's School in Birmingham. He was a keen chess-player and by the age of fourteen, he was Midlands under 21 Chess Champion. As a child he was exposed to quite a lot of mathematics, including Coxeter's polyhedron paper and Cundy and Rollett's

*Mathematical models*.

Borcherds entered Trinity College, Cambridge where he was an undergraduate. After the award of his B.A., he proceeded to undertake research supervised by John Conway. At first he felt that he did not have what it takes to be a research mathematician [8]:-

I wasn't getting very far. Most of the time I was struggling to keep my job. I'd see other people my age, such as Simon Donaldson (1986 Fields Medallist), being considerably more successful, and I thought I'm obviously not all that good. There were times when I thought of dropping out.However, he was highly successful and was awarded his doctorate in 1985 for his thesis

*The Leech lattice and other lattices*. The Leech lattice, discovered by John Leech in 1965, gave a dense packing of spheres in 24 dimensions. From the automorphism group of this lattice, Conway had discovered three previously unknown finite simple groups in 1968. Borcherds wrote in the Preface to his thesis:-

I thank my research supervisor Professor J H Conway for his help and encouragement. I also thank the S.E.R.C. for its financial support and Trinity College for a research scholarship and a fellowship.He published a paper

*The Leech lattice*in 1985 and

*Vertex algebras, Kac-Moody algebras and the monster*in the following year. He had invented to the idea of a vertex algebra which has proved to be extremely significant but at first few realised its importance [8]:-

I was pretty pleased with it at the time but after a few years I got a bit disillusioned, because it was obvious that nobody else was really interested in it. There is no point in having an idea that is so complicated that nobody can understand it. I remember I used to give talks on vertex algebras, and usually nobody turned up. Then there was this one time when I got a really big audience. But there had been a misprint, and the title read "vortex algebras", not "vertex algebras". The audience was made up of fluid physicists, and when they realised it was a misprint, they weren't interested either in what I had to say.Borcherds had been appointed as a Research Fellow at Trinity College, Cambridge, in 1983 and held this post until 1987. He then spent the academic year 1987-88 as Morrey Assistant Professor at the University of California, Berkeley, before returning to Cambridge where he was appointed as a Royal Society University Research Fellow in 1988. His most famous result was to prove the "moonshine conjecture" in the spring of 1989. He had spent eight years thinking about this conjecture while producing many other highly significant results. He had the inspiration necessary to prove the conjecture while travelling in India [8]:-

I was in Kashmir. I had been travelling around northern India, and there was one really long tiresome bus journey, which lasted about 24 hours. Then the bus had to stop because there was a landslide and we couldn't go any further. It was all pretty darn unpleasant. Anyway, I was just toying with some calculations on this bus journey and finally I found an idea which made everything work.The ideas behind the moonshine conjecture are hard to explain. Here is what Allyn Jackson writes in [5]:-

In the classification of finite simple groups, one of the most mysterious objects found was the monster group. There are various conjectures that attempt to connect the monster to other parts of mathematics. Borcherds invented the notion of a vertex algebra and used it to solve the Conway-Norton conjecture, which concerns the representation theory of the monster group (this theory is sometimes called "monstrous moonshine"). He used these results to generate product formulae for certain modular and automorphic forms. The first such formulae were found in the one-dimensional case by Euler and Jacobi, and the conventional wisdom in algebraic geometry was that such product formulae could not exist in higher dimensions. Borcherds's work is also important in physics, as it lays rigorous groundwork for conformal field theory in two dimensions.Borcherds remained as a Royal Society University Research Fellow at Cambridge until 1992, and then spent the following year as a lecturer at Cambridge. In 1993 he was appointed as Professor of Mathematics at the University of California, Berkeley. He returned to Cambridge in 1996 and spent three years there as a Royal Society Professor in the Department of Mathematics before returning to his professorship at the University of California, Berkeley, in 1999. He continues to hold this position.

In 1992 Borcherds was awarded a Junior Whitehead Prize by the London Mathematical Society. The citation reads:-

A Junior Whitehead Prize 1992 is awarded to Dr Richard Borcherds of Cambridge University for his work on mathematical aspects of conformal field theory. Borcherds has made a number of important and original contributions to mathematics, with striking and surprising applications. One notable piece of work is his generalisation of the Kac-Moody algebras, to which much of classical representation theory applies. For one of the algebras in his class, the analogue of the Weyl Denominator Formula gives a remarkable, and quite unexpected, factorisation of the classical modular function as an infinite product. Another important concept invented by Borcherds is that of a 'vertex operator algebra'. This is a way of formulating the essential mathematical ideas of conformal quantum field theory. The concept was used extensively by Frenkel, Lepowski and Meurman in their work on the Monster simple group, and very recently it has been an essential tool in the progress made by E Frenkel on a conjecture of Drinfeld about the Langlands correspondence for representations of loop groups. In a marvellous synthesis of the ideas which he has been developing, Borcherds has proved conjectures of Conway and Norton on the 'Moonshine' module for the Monster group.Also in 1992 he received a prize from the European Mathematical Society at the European Congress of Mathematicians in Paris. On 10 March 1994 he was elected to a fellowship of the Royal Society. His greatest honour, however, was being awarded a Fields Medal on 18 August 1998 at the Opening Ceremonies of the International Congress of Mathematicians in Berlin. The award was made:-

... for his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie Algebras, the proof of the Conway-Norton moonshine conjecture and the discovery of a new class of automorphic infinite products.However, Borcherds took the fame which went with this honour in his stride [8]:-

Before the award I used to think it was terribly important, but now I realise that it's meaningless. However, I was over the moon when I proved the moonshine conjecture. If I get a good result I spend several days feeling really happy about it. I sometimes wonder if this is the feeling you get when you take certain drugs. I don't actually know, as I have not tested this theory of mine.Borcherds is married to the topologist Ursula Gritsch and has two daughters. His current research interests include a mathematically rigorous approach to quantum field theory.

### References (show)

- R E Borcherds, What is .. The Monster,
*Notices Amer. Math. Soc.***49**(9) (2002), 1076-1077. - Fields Medal Prize Winners (1998): Richard Ewen Borcherds (born 29 November 1959). http://www.icm2002.org.cn/general/prize/medal/1998.htm
- P Goddard, The work of Richard Ewen Borcherds,
*Proceedings of the International Congress of Mathematicians*I (Berlin, 1998), 99-108. - K Gold, The high-flying obsessives,
*The Guardian*(12 December 2000). - A Jackson, Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals,
*Notices Amer. Math. Soc.***45**(10) (1998), 1358-1360. - J Lepowsky, J Lindenstrauss, Y I Manin, and J Milnor, The Mathematical Work of the 1998 Fields Medalists,
*Notices Amer. Math. Soc.***46**(1) (1999), 17-26. - R Sanders, UC Berkeley professor wins highest honor in mathematics, the prestigious Fields Medal, University of California, Berkeley. http://berkeley.edu/news/media/releases/98legacy/08-19-1998a.html
- S Singh, Interview with Richard Borcherds,
*The Guardian*(28 August 1998).

### Additional Resources (show)

Other websites about Richard Borcherds:

### Honours (show)

Honours awarded to Richard Borcherds

Written by J J O'Connor and E F Robertson

Last Update July 2009

Last Update July 2009