**Efim Zelmanov** is a Russian-born American mathematician best known for his solution of the restricted Burnside problem.

- Zelmanov described this work on Jordan algebras in his invited lecture to the International Congress of Mathematicians at Warsaw in 1983.
- In 1980 Zelmanov was appointed as a Junior Researcher at the Institute of Mathematics of the USSR Academy of Sciences at Novosibirsk.
- In 1987 Zelmanov solved one of the big open questions in the theory of Lie algebras.
- This was a classical result for finite dimensional Lie algebras but Zelmanov solved a big open problem when he proved that the result also held for infinite dimensional Lie algebras.
- In 1990 Zelmanov was appointed a professor at the University of Wisconsin-Madison in the United States.
- The results mentioned above on Jordan algebras and Lie algebras would have guaranteed Zelmanov a place as one of the great algebraists of the 20th century.
- However, in 1991, Zelmanov went on to settle one of the most fundamental results in the theory of groups which had occupied group theorists throughout the 20th century.
- In 1994 Zelmanov was awarded a Fields Medal for this work at the International Congress of Mathematicians in Zürich in 1994.
- Let me explain the background to the restricted Burnside problem, the solution of which was the main reason for the award of the Medal, and also explain how Zelmanov, not a group theorist by training, came to solve one of the most fundamental questions in group theory.
- This Zelmanov achieved in 1989.
- Zelmanov next set about proving that a Lie algebra with an Engel condition was locally nilpotent.
- The proof uses a deep structure theory for (quadratic) Jordan algebras, previously developed by McCrimmon and Zelmanov, as well as divided powers and other tools; it also relies on the joint work of Kostrikin and Zelmanov, which establishes the local nilpotency of the so-called sandwich algebras.
- Filled with humour, they were all delivered with Zelmanov's infectious twinkle in his eyes.
- In addition to the Fields Medal, Zelmanov has received other honours for his outstanding work.
- Zelmanov held a professorship at Yale University from 1995 to 2002.
- Interestingly, Zelmanov occupied the same office at the University of California, San Diego, as had earlier been occupied by Fields Medal winners Shing-Tung Yau and Michael Freedman.
- Zelmanov's contribution to mathematics goes far beyond his remarkable research and teaching achievements, however, being an editor or on the editorial board of more than ten major mathematics journals, including The Annals of Mathematics, The Journal of Algebra and The Journal of the American Mathematical Society.

Born 7 September 1955, Khabarovsk, USSR (now Russia).

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Algebra, Prize Fields Medal, Group Theory, Origin Russia, Puzzles And Problems

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive