Richard Borcherds

fl. 1959, Cape Town, South Africa

AlgebraMathematical physicsFields Medal 1998

British mathematician who invented vertex operator algebras and generalized Kac–Moody algebras and used them to prove the Conway–Norton monstrous moonshine conjecture.

Richard Ewen Borcherds was born in Cape Town, South Africa, where his father was a physicist; the family returned to England and he was educated at King Edward's School, Birmingham and Trinity College, Cambridge. He completed his PhD in 1985 at Cambridge under John Conway. In the mid-1980s he introduced vertex operator algebras, building on the vertex operator construction in the representation theory of affine Lie algebras and on the Frenkel–Lepowsky–Meurman monster vertex algebra.

Borcherds then defined what are now called Borcherds algebras, or generalized Kac–Moody algebras, which extend the classical theory by allowing imaginary simple roots. In 1992 he used them to prove the Conway–Norton monstrous moonshine conjecture: the Fourier coefficients of the graded representations of the Monster group coincide with those of a distinguished family of modular functions. The proof, which produces the moonshine functions as an identity between two ways of computing the denominator of a monster Lie algebra, also led him to the theory of automorphic forms on orthogonal groups and the multiplicative 'Borcherds products'.

After a decade at Cambridge, where he became a Royal Society University Research Fellow and then lecturer, Borcherds moved to the University of California, Berkeley in 1993, becoming professor and later holding positions again at Cambridge as the Royal Society Research Professor and then the Rouse Ball Professor of Mathematics. He received the Fields Medal at the 1998 ICM in Berlin and was elected a Fellow of the Royal Society in 1994.

Workplaces: University of Cambridge, University of California, Berkeley

United Kingdom

Award-winning work

Fields Medal 1998Borcherds was awarded the medal for his proof of the Conway–Norton monstrous moonshine conjecture, linking the Monster simple group to modular functions. He introduced and developed two new structures: vertex operator algebras, which axiomatize the algebra of conformal field theory, and Borcherds algebras (generalized Kac–Moody algebras), which allow imaginary simple roots. Constructing a 'monster Lie algebra' acted on by the Monster group and computing its denominator identity, he proved that the graded dimensions of the monster vertex algebra are the monstrous moonshine functions predicted by Conway and Norton, and he discovered the automorphic infinite products now called Borcherds products.

Contributions, linked to the library

Library topicLie groups and Lie algebrasDefined generalized Kac–Moody algebras (Borcherds algebras), extending the Lie-algebraic theory behind affine Lie algebras.Library topicModular formsMoonshine connects the Monster's representations to modular functions; his Borcherds products are automorphic forms with known divisors.