Richard Borcherds
Richard Borcherds  

Born  Richard Ewen Borcherds 29 November 1959 ^{[1]} Cape Town, South Africa 
Residence  UK, US 
Nationality  British^{[2]} 
Fields  Mathematics 
Institutions  
Alma mater  Trinity College, Cambridge 
Thesis  The leech lattice and other lattices (1984) 
Doctoral advisor  John Horton Conway^{[3]} 
Doctoral students 

Known for  Borcherds algebra 
Notable awards 

Website math.berkeley.edu/~reb 
Richard Ewen Borcherds (/ˈbɔːrtʃərdz/; born 29 November 1959)^{[1]} is a BritishAmerican^{[4]} mathematician currently working in quantum field theory. He is known for his work in lattices, number theory, group theory, and infinitedimensional algebras,^{[3]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]} for which he was awarded the Fields Medal in 1998.
Early life
Borcherds was born in Cape Town, but the family moved to Birmingham in the United Kingdom when he was six months old.^{[17]} 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 chess after coming to believe that the higher levels of competitive chess are merely about the competition rather than the fun of playing.
Education
He was educated at King Edward's School, Birmingham and Trinity College, Cambridge,^{[18]} where he studied under John Horton Conway.^{[19]}
Career
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. He was a Royal Society University Research Fellow.^{[20]}^{[18]} From 1996 he held a Royal Society Research Professorship at Cambridge before returning to Berkeley in 1999 as Professor of mathematics.^{[18]} At Berkeley, he held a Miller Research Professorship from 2000 to 2001.
Borcherds's early work included pioneering results on classification of unimodular lattices, and the introduction of new algebraic objects, most notably vertex algebras and BorcherdsKacMoody algebras. These ideas came together in his vertexalgebraic construction and analysis of the fake monster Lie algebra (called the monster Lie algebra at the time).
Borcherds is best known for his resolution of the ConwayNorton monstrous moonshine conjecture, which describes an intricate relation between the monster group and modular functions on the complex upper halfplane. To prove this conjecture, he drew upon theories that he had previously introduced, namely those of vertex algebras and BorcherdsKacMoody algebras, together with techniques of string theory, and applied them to the "moonshine module", a vertex operator algebra with monster symmetry constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. Additional work in moonshine concerned mod p variants of this conjecture, and were known as modular moonshine.
Later contributions include the theory of Borcherds products, which are holomorphic automorphic forms on O(n,2) that have wellbehaved infinite product expansions at cusps. Borcherds used this theory to resolve some longstanding conjectures concerning quasiaffineness of certain moduli spaces of algebraic surfaces. More recently, Borcherds has rendered perturbative renormalization, in particular the 't HooftVeltman proof of perturbative renormalizability of gauge theory, into rigorous mathematical language.
An interview with Simon Singh for The Guardian, in which Borcherds suggested he might have some traits associated with Asperger syndrome,^{[17]} subsequently led to a chapter about him in a book on autism by Simon BaronCohen.^{[21]}^{[22]} BaronCohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.^{[21]}
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.^{[19]} In 1994, he was elected to the Royal Society of Fellows.^{[23]} 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.^{[19]} 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 ConwayNorton moonshine conjecture and the discovery of a new class of automorphic infinite products." In 2012 he became a fellow of the American Mathematical Society,^{[24]} and in 2014 he was elected to the National Academy of Sciences.^{[25]}
References
 ^ ^{a} ^{b} "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. ISSN 14310635. arXiv:math/9808136 ..
 ^ ^{a} ^{b} ^{c} Richard Borcherds at the Mathematics Genealogy Project
 ^ http://www.nasonline.org/memberdirectory/members/20033092.html
 ^ 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". International Mathematical Olympiad.
 ^ Richard Borcherds's publications indexed by the Scopus bibliographic database, a service provided by Elsevier. (subscription required)
 ^ 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:10.1016/00409383(95)000364.
 ^ Borcherds, R. E. (1986). "Vertex algebras, KacMoody algebras, and the Monster". Proceedings of the National Academy of Sciences of the United States of America. 83 (10): 3068–71. PMC 323452 . PMID 16593694. doi:10.1073/pnas.83.10.3068.
 ^ List of publications from Microsoft Academic Search
 ^ ^{a} ^{b} Simon Singh, "Interview with Richard Borcherds", The Guardian (28 August 1998)
 ^ ^{a} ^{b} ^{c} "UC Berkeley professor wins highest honor in mathematics, the prestigious Fields Medal". University of California, Berkeley. 19 August 1998. Retrieved 22 July 2009.

^ ^{a} ^{b} ^{c} "Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals" (PDF). Notices of the American Mathematical Society. American Mathematical Society. 45 (10).
first1=
missinglast1=
in Authors list (help)  ^ Cook, Alan (2000). "URFs become FRS: Frances Ashcroft, Athene Donald, and John Pethica". Notes and Records of the Royal Society. London: Royal Society. 54 (3): 409–411. doi:10.1098/rsnr.2000.0181.
 ^ ^{a} ^{b} BaronCohen, Simon (2004). "A Professor of Mathematics". The Essential Difference: Male and Female Brains and the Truth about Autism. Basic Books. ISBN 046500556X. (see external links) records conversations with Richard Borcherds and his family.
 ^ High flying obsessives, The Guardian, December 2000
 ^ "EC/1994/05: Borcherds, Richard Ewen". London: The Royal Society. Archived from the original on 27 May 2014.
 ^ List of Fellows of the American Mathematical Society. Retrieved 10 November 2012.
 ^ National Academy of Sciences Members and Foreign Associates Elected, National Academy of Sciences, 29 April 2014.
Further reading
 Conway and Sloane, Sphere Packings, Lattices, and Groups, Third Edition, Springer, 1998 ISBN 0387985859.
 Frenkel, Lepowsky and Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988 ISBN 0122670655.
 Kac, Victor, Vertex Algebras for Beginners, Second Edition, AMS 1997 ISBN 0821806432.
 20thcentury British mathematicians
 21stcentury British mathematicians
 Fields Medalists
 Group theorists
 Fellows of the Royal Society
 Fellows of the American Mathematical Society
 Members of the United States National Academy of Sciences
 University of California, Berkeley faculty
 1959 births
 Living people
 People educated at King Edward's School, Birmingham
 Alumni of Trinity College, Cambridge
 Whitehead Prize winners
 International Mathematical Olympiad participants
 British expatriate academics in the United States