Efim Zelmanov

Fields Medal


As a young mathematician in 1970s and 80s, Efim Zelmanov was well aware of the Burnside problem, which he says was “very popular” in the Soviet Union where he grew up. But never did Prof Zelmanov think he would one day be the person to solve the long-standing algebraic problem.

It was British mathematician William Burnside who, in 1902, first posed the question: does a finitely generated group, where every element has finite order, necessarily have to be considered a finite group? The question, which became known as the Burnside problem, is fundamental to group theory, the framework mathematicians use to study symmetries.

Prof Zelmanov was able to provide proof that certain algebraic constructs, known as periodic groups, are finite, and that there are limits to the number of symmetries an object can have. For his work, Prof Zelmanov was awarded the Fields Medal in 1994, the mathematical equivalent of winning the Nobel Prize. 

His work on the Burnside problem helped advance the field of algebra, opening up new areas of thinking and operations. Prof Zelmanov has also made important contributions to the theory of Lie algebras and Jordan algebras, which have impacted the study of geometry and quantum mechanics. The two subjects are deeply interconnected, he says. “Mathematics has always benefited a lot from physics, and physics has contributed a lot to mathematics.”

But what Prof Zelmanov loves about mathematics is the certainty it carries, compared with other subjects. “In physics, I thought I had solved the problem. But then some cleverer guy gives his solution and he’s right,” he jokes. “In mathematics, if it’s solved then it’s solved.”

Prof Zelmanov attributes his love for mathematics to a teacher he had when he was twelve. “She was not a great problem solver herself,” he recalls. “But she did everything right.” Generous with her praise and always encouraging, the teacher influenced several of her students, Prof Zelmanov included, to became professional mathematicians later in life.

Prof Zelmanov studied mathematics at the State University in Novosibirsk, Russia, where he obtained his Masters and PhD. Upon graduation, he became a researcher at the Soviet Institute for Mathematics before emigrating to the US in 1990 to take up a professorship at the University of Wisconsin-Madison.

Today, Prof Zelmanov teaches at the University of California San Diego. In 2001, at the age of 47, he was elected to the National Academy of Sciences, as the youngest member of the mathematics section. Prof Zelmanov is also a member of the American Academy of Arts and Sciences, the Korean Academy of Sciences and Technology, the Spanish Royal Academy of Sciences and the Brazilian Academy of Sciences. He sits on the editorial board of more than ten major mathematics journals.