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Richard Ewen Borcherds (; born 29 November 1959)^{[1]} is a British mathematician currently working in quantum field theory. He is known for his work in lattices, number theory, group theory, and infinite-dimensional algebras,^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]} for which 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.^{[16]} 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.
He was educated at King Edward's School, Birmingham and Trinity College, Cambridge,^{[17]} where he studied under John Horton Conway.^{[18]}
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.^{[17]} From 1996 he held a Royal Society Research Professorship at Cambridge before returning to Berkeley in 1999 as Professor of mathematics.^{[17]}
An interview with Simon Singh for the Guardian, in which Borcherds suggested he might have some traits associated with Asperger syndrome,^{[16]} subsequently led to a chapter about him in a book on autism by Simon Baron-Cohen.^{[19]}^{[20]} Baron-Cohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.^{[19]}
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.
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.^{[18]} 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.^{[18]} 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,^{[21]} and in 2014 he was elected to the National Academy of Sciences.^{[22]}
His nomination for the Royal Society reads:
Isle of Man, India, Canada, European Union, British Overseas Territories
Quantum mechanics, Standard Model, Condensed matter physics, Quantum electrodynamics, Quantum chromodynamics
Logic, Set theory, Statistics, Number theory, Mathematical logic
University of Cambridge, Institute for Advanced Study, Princeton University, University of California, Berkeley, University of Oxford
South Africa, Table Mountain, Nice, Western Cape, Madagascar
String theory, Conformal field theory, Mathematics, Quantum gravity, Vertex operator algebra
Number theory, Abstract algebra, Simple group, Hydrogen, Abelian group
University of Texas at Austin, University of California, Berkeley, University of Utah, Group theory, Algebraic geometry
Conway's Game of Life, Liverpool, Princeton University, Gonville and Caius College, Cambridge, University of Cambridge