The thesis he presented for his Ph.D. was on nonassociative algebra. In particular his work completely changed the whole of the subject of Jordan algebras by extending results from the classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras. 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. On the award of his doctorate (habilitation) in 1985, he was promoted to Senior Researcher. He was promoted again at the Institute of Mathematics of the USSR Academy of Sciences in 1986, this time becoming a Leading Researcher.
In 1987 Zelmanov solved one of the big open questions in the theory of Lie algebras. He proved that the Engel identity
In 1990 Zelmanov was appointed a professor at the University of Wisconsin-Madison in the United States. He held this appointment until 1994 when he was appointed to the University of Chicago. In 1995 he spent the year at Yale University.
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. He solved the restricted Burnside problem.
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.
In 1902 Burnside first asked whether a finitely generated group in which every element has finite order, is finite. This problem is known as the General Burnside problem. The Burnside problem asks whether, for fixed d and n, the group B(d, n) having d generators and in which every element satisfies xn = 1, is finite. It is really easy to show the B(d, 2) is finite. Burnside himself showed that B(d, 3) is finite, Sanov showed B(d, 4) is finite and Marshall Hall showed B(d, 6) is finite.
By the 1930s no real progress had been made on either of these problems and the Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for fixed d and n, there is a largest finite d generator group in which every element satisfies xn = 1. This is equivalent to saying that a positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of B(d, n).
The General Burnside problem was shown to have a negative solution by Golod in 1964. In 1968 Novikov and Adian showed that the Burnside problem was false for large n. The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime powers n. The Schreier conjecture, that the outer automorphism groups of finite simple groups are soluble, was shown to be true as a consequence of the classification of finite simple groups.
Magnus had reduced the case of the Restricted Burnside problem for n prime to a question about whether Lie algebras satisfying an Engel condition are locally nilpotent. Kostrikin, in 1959, proved that such Lie algebras were indeed locally nilpotent. However Kostrikin's proof is not entirely satisfactory and a corrected version only appeared much later.
When Zelmanov began to work on the Restricted Burnside problem there were two major difficulties in pushing what had been achieved for n = p to n = pk. Firstly it there was no reduction of the problem to Lie algebras with the Engel condition, This Zelmanov achieved in 1989.
Zelmanov next set about proving that a Lie algebra with an Engel condition was locally nilpotent. This he achieved in two papers, the first dealing with odd prime characteristic and the second dealing with n = pk which corresponds to Lie algebras of characteristic 2. Shalev writes in :-
His stunning proof ... combines an amazing technical capability with highly original ideas from various disciplines. 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. While Lie algebras have long been considered a natural playground in the context of the Restricted Burnside problem, the appearance of Jordan algebras is unprecedented and quite surprising.At the Groups-St Andrews conference at Galway, Ireland in 1993, of which I [EFR] was a joint organiser, Zelmanov was one of the main speakers and he gave a series of five lectures on Nil rings methods in the theory of nilpotent groups. His lectures were beautifully constructed, models of clarity, showing what had been achieved and presenting many glimpses of possible directions for future research. 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. He received the Collège de France Medal in January 1992 and the Andre Aizenstadt Prize in May 1996.
Zelmanov held a professorship at Yale University from 1995 to 2002. During his last year at Yale he was elected to the National Academy of Sciences, becoming the youngest member in the Academy's mathematics division. In 2002 he left Yale and went to the University of California, San Diego, where he was appointed to the Rita L Atkinson Endowed Chair in Mathematics. At the time of his appointment, James Bunch, chair of mathematics at the University of California, San Diego, said:-
Professor Zelmanov is one of the top mathematicians in the world and he will play an important role in furthering the international reputation and tradition of excellence of UCSD's mathematics department. He is also an outstanding teacher and I expect him to be an exceptional role model for our students.Jeffrey Remmel, associate dean of the UCSD's Division of Physical Sciences and the former chair of mathematics who recruited Zelmanov to UCSD said:-
Professor Zelmanov's presence at UCSD ensures that we have one of the leading research groups in algebra and representation theory in the country. In addition, he is a superb lecturer and thesis advisor. He will help the mathematics department attract the best young researchers and graduate students in the field of algebra and will have a profound effect on the next generation of mathematics students at UCSD.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.
Article by: J J O'Connor and E F Robertson