Efim Zelmanov

Native spelling: Ефим Зельманов

fl. 1955, Khabarovsk, Soviet Union

AlgebraCombinatorics and discrete mathematicsFields Medal 1994

Russian-American algebraist who solved the restricted Burnside problem on groups of finite exponent and transformed the theory of Jordan and Lie algebras.

Efim Isaakovich Zelmanov was born in Khabarovsk in the Soviet Far East and grew up in Novosibirsk, where he attended the famed physics and mathematics boarding school and then Novosibirsk State University. He earned his candidate of sciences degree in 1980 and his doctorate in 1985 at the Institute of Mathematics of the Siberian Branch of the Soviet Academy of Sciences, working under Anatoly Shirshov on non-associative algebra. His early work included classification results on Jordan algebras and superalgebras that later proved essential in his group-theoretic breakthroughs.

In 1987 Zelmanov began spending extended periods abroad, and in 1990 he moved permanently to the West, holding professorships at the University of Wisconsin–Madison, the University of Chicago, and Yale University, before joining the University of California, San Diego in 2002. In 1990–1991 he announced and published his solution of the restricted Burnside problem, a question posed in the 1930s and sharpened by Kostrikin and others: whether there is a largest finite quotient of a free Burnside group of fixed exponent on a fixed number of generators. By translating the question into the theory of Lie algebras with an Engel condition and applying deep results on Jordan structures, he proved that such a largest finite group always exists.

Zelmanov received the Fields Medal at the 1994 International Congress of Mathematicians in Zurich. He has since continued to work on Lie algebras, Jordan algebras, Kac–Moody algebras and related areas, and has also been active in the organization of mathematics, serving as a member of the Fields Medal Committee and of international prize juries. In 2022 he moved to the Korea Institute for Advanced Study in Seoul.

Workplaces: Institute of Mathematics, Siberian Branch of the Academy of Sciences (Novosibirsk), University of Chicago, Yale University, Korea Institute for Advanced Study (KIAS)

Russia, United States

Award-winning work

Fields Medal 1994Zelmanov received the medal for his solution of the restricted Burnside problem: he proved that for every pair of positive integers d and n there exists a largest finite group that is generated by d elements and in which every element has order dividing n, settling a question about periodic groups open since the 1950s. His proof recast the group-theoretic problem as one about Lie algebras satisfying an Engel-type condition, and it drew on the theory of Jordan algebras and superalgebras, whose simple infinite-dimensional analogues he had helped classify. The result completed the Burnside program in the restricted case and transformed the study of groups of finite exponent and of non-associative algebras.

Contributions, linked to the library

Great problemBurnside problemSolved the restricted Burnside problem (1991): a largest finite quotient exists for every exponent and number of generators.Library topicLie groups and Lie algebrasHis proof hinged on Lie algebras satisfying an Engel condition; he also worked on Lie superalgebras and Kac–Moody algebras.