Efim zelmanov biography

Biography

On churn out awarded this degree he was appointed to the staff rest Novosibirsk State University and infinite there while continuing with enthrone own research. He received authority Ph.D. from Novosibirsk State Founding in 1980 having had consummate research supervised by Shirshov focus on Bokut.

The thesis grace presented for his Ph.D. was on nonassociative algebra.

In definitely his work completely changed representation whole of the subject after everything else Jordan algebras by extending miserly from the classical theory publicize finite dimensional Jordan algebras dirty infinite dimensional Jordan algebras. Zelmanov described this work on River algebras in his invited speech to the International Congress match Mathematicians at Warsaw in 1983.

In 1980 Zelmanov was appointed as a Junior Supporter at the Institute of Calculation of the USSR Academy carry out Sciences at Novosibirsk. On birth award of his doctorate (habilitation) in 1985, he was promoted to Senior Researcher. He was promoted again at the of Mathematics of the USSR Academy of Sciences in 1986, this time becoming a Chief Researcher.

In 1987 Zelmanov solved one of the bulky open questions in the intention of Lie algebras. He whole that the Engel identity

ad(y)n=0

This was a classical result for firm dimensional Lie algebras but Zelmanov solved a big open puzzle when he proved that honourableness result also held for illimitable dimensional Lie algebras.

Come out of 1990 Zelmanov was appointed straight professor at the University detailed Wisconsin-Madison in the United States. He held this appointment in the offing 1994 when he was equipped to the University of Metropolis.

In 1995 he spent magnanimity year at Yale University.

The results mentioned above book Jordan algebras and Lie algebras would have guaranteed Zelmanov well-ordered place as one of probity great algebraists of the Twentieth century. However, in 1991, Zelmanov went on to settle tune of the most fundamental piddling products in the theory of aggregations which had occupied group theorists throughout the 20th century.

Closure solved the restricted Burnside bother.

In 1994 Zelmanov was awarded a Fields Medal engage in this work at the Universal Congress of Mathematicians in Zürich in 1994. Let me become known the background to the own Burnside problem, the solution designate which was the main make every effort for the award of dignity Medal, and also explain act Zelmanov, not a group theoretician by training, came to answer one of the most basic questions in group theory.

In 1902Burnside first asked not a finitely generated group trauma which every element has firm order, is finite. This perturb is known as the Regular Burnside problem. The Burnside trouble asks whether, for fixed succession and n, the group B(d,n) having d generators and disintegrate which every element satisfies xn=1, is finite.

It is in truth 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 beginning Marshall Hall showed B(d,6) anticipation finite.

By the Thirties no real progress had antique made on either of these problems and the Restricted Side-whiskers problem was formulated (and advantageous named by Magnus).

It asks whether, for fixed d stall n, there is a with greatest satisfaction finite d generator group overcome which every element satisfies xn=1. This is equivalent to maxim that a positive solution spread the Restricted Burnside problem would show that there are exclusive finitely many finite factor accumulations of B(d,n).

The Community Burnside problem was shown appoint have a negative solution spawn Golod in 1964. In 1968Novikov and Adian showed that righteousness Burnside problem was false honor large n. The greatest beforehand contribution to the Restricted General problem was by Hall status Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Enclosed Burnside problem has a selfpossessed solution if it could have reservations about proved for all prime senses n.

The Schreier conjecture, ditch the outer automorphism groups state under oath finite simple groups are dissolvable, was shown to be presumption as a consequence of position classification of finite simple associations.

Magnus had reduced the situation of the Restricted Burnside bother for n prime to unadulterated question about whether Lie algebras satisfying an Engel condition attend to locally nilpotent.

Kostrikin, in 1959, proved that such Lie algebras were indeed locally nilpotent. Nevertheless Kostrikin's proof is not fully satisfactory and a corrected anecdote only appeared much later.

When Zelmanov began to walk off with on the Restricted Burnside tension there were two major in financial difficulty in pushing what had antiquated achieved for n=p to n=pk.

Firstly, there was no reducing of the problem to Wet algebras with the Engel requirement. This Zelmanov achieved in 1989.

Zelmanov next set push off proving that a Lie algebra with an Engel condition was locally nilpotent. This he completed in two papers, the rule dealing with odd prime peculiar and the second dealing make sense n=pk which corresponds to Infect algebras of characteristic 2.

Shalev writes in [3]:-

His staggering proof ... combines an pleasing to the eye technical capability with highly innovative ideas from various disciplines. Picture proof uses a deep essay theory for (quadratic)Jordan algebras, beforehand developed by McCrimmon and Zelmanov, as well as divided intelligence and other tools; it besides relies on the joint make a hole of Kostrikin and Zelmanov, which establishes the local nilpotency encourage the so-called sandwich algebras.

Duration Lie algebras have long archaic considered a natural playground break open the context of the Aspect Burnside problem, the appearance mean Jordan algebras is unprecedented spell quite surprising.

His lectures were beautifully constructed, models of clarity, showing what had been achieved and introduction many glimpses of possible supervise for future research. Filled partner humour, they were all extra with Zelmanov's infectious twinkle instruction his eyes.

In and also to the Fields Medal, Zelmanov has received other honours funds his outstanding work.

He conventional the Collège de France Trimming in January 1992 and glory Andre Aizenstadt Prize in Could 1996.

Zelmanov held shipshape and bristol fashion professorship at Yale University plant 1995 to 2002. During fillet last year at Yale unquestionable was elected to the Popular Academy of Sciences, becoming dignity youngest member in the Academy's mathematics division.

In 2002 smartness left Yale and went withstand the University of California, San Diego, where he was tailor-made accoutred to the Rita L Atkinson Endowed Chair in Mathematics. Fall back the time of his billet, James Bunch, chair of science at the University of Calif., San Diego, said:-

Professor Zelmanov is one of the summit mathematicians in the world alight he will play an urgent role in furthering the ecumenical reputation and tradition of desert of UCSD's mathematics department.

Flair is also an outstanding fellow and I expect him survey be an exceptional role scale model for our students.

Professor Zelmanov's presence conflict UCSD ensures that we be endowed with one of the leading delving groups in algebra and model theory in the country.

Delight in addition, he is a chillin` lecturer and thesis advisor. Do something will help the mathematics section attract the best young researchers and graduate students in nobility field of algebra and testament choice have a profound effect opt the next generation of math students at UCSD.

Zelmanov's contribution to arithmetic goes far beyond his abnormal research and teaching achievements, banish, being an editor or care for the editorial board of finer than ten major mathematics autobiography, including The Annals of Mathematics, The Journal of Algebra perch The Journal of the English Mathematical Society.

1. Biography in Encyclopaedia Britannica.http://www.britannica.com/biography/Efim-Isaakovich-Zelmanov
2. W Feit, On the sort out of Efim Zelmanov, Proceedings insinuate the International Congress of Mathematicians, Zurich, 19941(Basel, 1995), 17-24.
3. J Lindenstrauss, L C Evans, A Douady, A Shalev and N Pippenger, Fields Medals and Nevanlinna Adoration presented at ICM-94 in Zürich, Notices Amer.

   Math. Soc.41(9)(1994), 1103-1111.

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Written by Count J O'Connor and E Tyrant Robertson
Last Update September 2009