|Born||Richard Ewen Borcherds
29 November 1959 
Cape Town, South Africa
|Alma mater||Trinity College, Cambridge|
|Thesis||The leech lattice and other lattices (1984)|
|Doctoral advisor||John Horton Conway|
|Known for||Borcherds algebra|
Richard Ewen Borcherds (born 29 November 1959) is a British mathematician specializing in lattices, number theory, group theory, and infinite-dimensional algebras. He was awarded the Fields Medal in 1998.
Borcherds was born in Cape Town, but the family moved to Birmingham in the United Kingdom when he was six months old. His father is a physicist and he has three brothers, two of whom are mathematics teachers. He was a promising mathematician and chess player as a child, winning several national mathematics championships and "was in line for becoming a chess master" before giving up after coming to believe that the higher levels of competitive chess are merely about the competition rather than the fun of playing.
After receiving his doctorate in 1985 he has held various alternating positions at Cambridge and the University of California, Berkeley, serving as Morrey Assistant Professor of Mathematics at Berkeley from 1987 to 1988. From 1996 he held a Royal Society Research Professorship at Cambridge before returning to Berkeley in 1999 as Professor of mathematics.
An interview with Simon Singh for the Guardian, in which Borcherds suggested he might have some traits associated with Asperger syndrome, subsequently led to a chapter about him in a book on autism by Simon Baron-Cohen. Baron-Cohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.
Borcherds is best known for his work connecting the theory of finite groups with other areas in mathematics. In particular he invented the notion of vertex algebras, which Igor Frenkel, James Lepowsky and Arne Meurman used to construct an infinite-dimensional graded algebra acted on by the monster group. Borcherds then used this, and methods from string theory, to prove the monstrous moonshine conjecture by Conway and Norton, relating the monster group to the coefficients of the q-expansion of the j-invariant. The result was not only a great increase in understanding of the monster group, a very large finite simple group whose structure was previously not well understood, but tied the monster to various aspects of mathematics and mathematical physics. In recent years, Borcherds has been attempting to construct quantum field theory in a mathematically rigorous manner.
Awards and honours
In 1992 he was one of the first recipients of the EMS prizes awarded at the first European Congress of Mathematics in Paris, and in 1994 he was an Invited Speaker at the International Congress of Mathematicians in Zurich. In 1998 at the 23rd International Congress of Mathematicians in Berlin, Germany he received the Fields Medal together with Maxim Kontsevich, William Timothy Gowers and Curtis T. McMullen. The award cited him "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." In 2012 he became a fellow of the American Mathematical Society, and in 2014 he was elected to the National Academy of Sciences.
His nomination for the Royal Society reads:
|“||Borcherds is distinguished for his seminal work applying the ideas of conformal quantum field theory to solve classical problems in pure mathematics. At the same time his work has led to a deeper understanding of conformal field theory. After important work on the classification of lattices, Borcherds introduced the notions of a vertex algebra, which played a key role in the construction of the natural representation of the monster group (the largest sporadic finite simple group) and a generalized Kac–Moody algebra, which led him to a proof of the moonshine conjectures of Conway and Norton. The character formula he proved for such algebras, together with the observation that their denominator functions are often automorphic forms on 0(n,2) led him to a remarkable product formula for the modular function j, amongst other results. Recently, his introduction of quantum rings, generalising vertex algebras, has provided a natural setting for the elusive concept of fusion in conformal field theory.||”|
- "BORCHERDS, Prof. Richard Ewen". Who's Who 2014, A & C Black, an imprint of Bloomsbury Publishing plc, 2014; online edn, Oxford University Press.(subscription required)
- Goddard, Peter (1998). The work of Richard Ewen Borcherds. "Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)". Documenta Mathematica. pp. 99–108. arXiv:math/9808136. ISSN 1431-0635..
- Richard Borcherds at the Mathematics Genealogy Project
- James Lepowsky, "The Work of Richard Borcherds", Notices of the American Mathematical Society, Volume 46, Number 1 (January 1999).
- Richard Borcherds, "What is ... The Monster?", Notices of the American Mathematical Society, Volume 49, Number 9 (October 2002).
- Richard Borcherds' web site (has links to some relatively informal lecture notes describing his work)
- O'Connor, John J.; Robertson, Edmund F., "Richard Borcherds", MacTutor History of Mathematics archive, University of St Andrews.
- Richard Borcherds's results at the International Mathematical Olympiad
- Richard Borcherds from the Scopus bibliographic database.
- Borcherds, R. E. (1992). "Monstrous moonshine and monstrous Lie superalgebras". Inventiones Mathematicae 109: 405. doi:10.1007/BF01232032.
- Borcherds, R. E. (1995). "Automorphic forms onO s +2,2(R) and infinite products". Inventiones Mathematicae 120: 161. doi:10.1007/BF01241126.
- Borcherds, R. E. (1998). "Automorphic forms with singularities on Grassmannians". Inventiones Mathematicae 132 (3): 491. doi:10.1007/s002220050232.
- Borcherds, R. E. (1996). "The Moduli space of Enriques surfaces and the fake monster lie superalgebra". Topology 35 (3): 699. doi:10.1016/0040-9383(95)00036-4.
- Borcherds, R. E. (1986). "Vertex algebras, Kac-Moody algebras, and the Monster". Proceedings of the National Academy of Sciences of the United States of America 83 (10): 3068–71. doi:10.1073/pnas.83.10.3068. PMC 323452. PMID 16593694.
- List of publications from Microsoft Academic Search
- Simon Singh, "Interview with Richard Borcherds", The Guardian (28 August 1998)
- "UC Berkeley professor wins highest honor in mathematics, the prestigious Fields Medal". University of California, Berkeley. 19 August 1998. Retrieved 2009-07-22.
- "Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals". Notices of the American Mathematical Society (American Mathematical Society) 45 (10).
- Baron-Cohen, Simon (2004). The Essential Difference: Male and Female Brains and the Truth about Autism. Basic Books. ISBN 0-465-00556-X.. Chapter 11, "A Professor of Mathematics" (see external links) records conversations with Richard Borcherds and his family.
- High flying obsessives, The Guardian, December 2000
- List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
- National Academy of Sciences Members and Foreign Associates Elected, National Academy of Sciences, April 29, 2014.
- "EC/1994/05: Borcherds, Richard Ewen". London: The Royal Society. Archived from the original on 2014-05-27.
- Conway and Sloane, Sphere Packings, Lattices, and Groups, Third Edition, Springer, 1998 ISBN 0-387-98585-9.
- Frenkel, Lepowsky and Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988 ISBN 0-12-267065-5.
- Kac, Victor, Vertex Algebras for Beginners, Second Edition, AMS 1997 ISBN 0-8218-0643-2.