Richard Borcherds
Encyclopedia
Richard Ewen Borcherds is a British
mathematician
specializing in lattices
, number theory
, group theory
, and infinite-dimensional algebras. He was awarded the Fields Medal
in 1998.
, 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. He was educated at King Edward's School, Birmingham
and Trinity College, Cambridge
, where he studied under John Horton Conway
. 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.
s 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.
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."
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
specializing in lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
, number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, and infinite-dimensional algebras. He was awarded the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1998.
Personal life
Borcherds was born in Cape TownCape Town
Cape Town is the second-most populous city in South Africa, and the provincial capital and primate city of the Western Cape. As the seat of the National Parliament, it is also the legislative capital of the country. It forms part of the City of Cape Town metropolitan municipality...
, but the family moved to Birmingham
Birmingham
Birmingham is a city and metropolitan borough in the West Midlands of England. It is the most populous British city outside the capital London, with a population of 1,036,900 , and lies at the heart of the West Midlands conurbation, the second most populous urban area in the United Kingdom with a...
in the United Kingdom
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
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
Chess master
A chess master is a chess player of such skill that he/she can usually beat chess experts, who themselves typically prevail against most amateurs. Among chess players, the term is often abbreviated to master, the meaning being clear from context....
" 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. He was educated at King Edward's School, Birmingham
King Edward's School, Birmingham
King Edward's School is an independent secondary school in Birmingham, England, founded by King Edward VI in 1552. It is part of the Foundation of the Schools of King Edward VI in Birmingham, and is widely regarded as one of the most academically successful schools in the country, according to...
and Trinity College, Cambridge
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
, where he studied under John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
. After receiving his doctorate in 1985 he has held various alternating positions at Cambridge and the University of California, Berkeley
University of California, Berkeley
The University of California, Berkeley , is a teaching and research university established in 1868 and located in Berkeley, California, USA...
, serving as Morrey Assistant Professor of Mathematics at Berkeley from 1987 to 1988. From 1996 he held a Royal Society Research Professorship
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
at Cambridge before returning to Berkeley in 1999 as Professor of mathematics.
An interview with Simon Singh
Simon Singh
Simon Lehna Singh, MBE is a British author who has specialised in writing about mathematical and scientific topics in an accessible manner....
for the Guardian
The Guardian
The Guardian, formerly known as The Manchester Guardian , is a British national daily newspaper in the Berliner format...
, in which Borcherds suggested he might have some traits associated with Asperger syndrome
Asperger syndrome
Asperger's syndrome that is characterized by significant difficulties in social interaction, alongside restricted and repetitive patterns of behavior and interests. It differs from other autism spectrum disorders by its relative preservation of linguistic and cognitive development...
, subsequently led to a chapter about him in a book on autism by Simon Baron-Cohen
Simon Baron-Cohen
Simon Baron-Cohen FBA is professor of Developmental Psychopathology in the Departments of Psychiatry and Experimental Psychology at the University of Cambridge in the United Kingdom. He is the Director of the University's Autism Research Centre, and a Fellow of Trinity College...
. Baron-Cohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.
Work
Borcherds is best known for his work connecting the theory of finite groupFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s with other areas in mathematics. In particular he invented the notion of vertex algebras, which Igor Frenkel
Igor Frenkel
Igor Borisovich Frenkel is a mathematician at Yale University working in representation theory and mathematical physics.-Biography:...
, James Lepowsky
James Lepowsky
James Lepowsky is a professor of mathematics at Rutgers University, New Jersey. Previously he taught at Yale University. He received his Ph.D from M.I.T. in 1970 where his advisor was Bertram Kostant. Lepowsky graduated from Stuyvesant High School in 1961, 16 years after Kostant...
and Arne Meurman
Arne Meurman
Arne Meurman is a Swedish mathematician working on finite groups and vertex operator algebras. Currently he is a professor at Lund University....
used to construct an infinite-dimensional graded algebra acted on by the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
. Borcherds then used this, and methods from string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, to prove the monstrous moonshine conjecture
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
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
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
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
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
in a mathematically rigorous manner.
Awards
In 1992 he was one of the first recipients of the EMS prizesEuropean Mathematical Society
The European Mathematical Society is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians...
awarded at the first European Congress of Mathematics
European Congress of Mathematics
The European Congress of Mathematics is an international congress of the mathematics community, held every four years. Its objectives are "to present various new aspects of pure and applied mathematics to a wide audience, to be a forum for discussion of the relationship between mathematics and...
in Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
, and in 1994 he was an Invited Speaker at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Zurich. In 1998 at the 23rd International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, Germany he received the Fields Medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
together with Maxim Kontsevich
Maxim Kontsevich
Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...
, William Timothy Gowers
William Timothy Gowers
William Timothy Gowers FRS is a British mathematician. He is a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at Cambridge University, where he also holds the Rouse Ball chair, and is a Fellow of Trinity College...
and Curtis T. McMullen
Curtis T. McMullen
Curtis Tracy McMullen is Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory....
. 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."
Sources
- Conway and SloaneNeil SloaneNeil James Alexander Sloane is a British-U.S. mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing...
, 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, VictorVictor KacVictor G. Kac is a Soviet and American mathematician at MIT, known for his work in representation theory. He discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities...
, Vertex Algebras for Beginners, Second Edition, AMS 1997 ISBN 0-8218-0643-2.
External links
- 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)