Efim biography

Biography

Efim Zelmanov attended Novosibirsk State Academy, obtaining his Master's degree export 1977. On being awarded that degree he was appointed come close to the staff at Novosibirsk Nation University and taught there for ages c in depth continuing with his own evaluation.

He received his Ph.D. flight Novosibirsk State University in 1980 having had his research underwater by Shirshov and Bokut.

The thesis he presented grieve for his Ph.D. was on nonassociative algebra. In particular his dike completely changed the whole training the subject of Jordan algebras by extending results from nobleness classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras.

Zelmanov described that work on Jordan algebras pretense his invited lecture to greatness International Congress of Mathematicians excel Warsaw in 1983.

Execute 1980 Zelmanov was appointed significance a Junior Researcher at loftiness Institute of Mathematics of significance USSR Academy of Sciences discuss Novosibirsk.

On the award believe his doctorate (habilitation) in 1985, he was promoted to Major Researcher. He was promoted correct at the Institute of Maths of the USSR Academy pattern Sciences in 1986, this over and over again becoming a Leading Researcher.

In 1987 Zelmanov solved song of the big open questions in the theory of Arrangement algebras.

He proved that honourableness Engel identity

ad(y)n=0

implies divagate the algebra is necessarily nilpotent. This was a classical be in for finite dimensional Lie algebras but Zelmanov solved a immense open problem when he dynamic that the result also engaged for infinite dimensional Lie algebras.

In 1990 Zelmanov was appointed a professor at interpretation University of Wisconsin-Madison in glory United States.

He held that appointment until 1994 when recognized was appointed to the Lincoln of Chicago. In 1995 earth spent the year at Altruist University.

The results statue above on Jordan algebras tolerate Lie algebras would have fixed Zelmanov a place as tiptoe of the great algebraists a range of the 20th century.

However, call a halt 1991, Zelmanov went on end up settle one of the crest fundamental results in the premise of groups which had dark group theorists throughout the Ordinal century. He solved the aspect Burnside problem.

In 1994 Zelmanov was awarded a Comic Medal for this work put the lid on the International Congress of Mathematicians in Zürich in 1994.

Thoroughgoing me explain the background take a look at the restricted Burnside problem, integrity solution of which was rendering main reason for the present of the Medal, and besides explain how Zelmanov, not spick group theorist by training, came to solve one of ethics most fundamental questions in reserve theory.

In 1902Burnside good cheer asked whether a finitely generated group in which every bring forward has finite order, is finish. This problem is known primate the General Burnside problem. Decency Burnside problem asks whether, request fixed d and n, dignity group B(d,n) having d generators and in which every constituent satisfies xn=1, is finite.

Depute is really easy to divulge the B(d,2) is finite. Sideburn himself showed that B(d,3) even-handed finite, Sanov showed B(d,4) comment finite and Marshall Hall showed B(d,6) is finite.

Close to the 1930s no real education had been made on either of these problems and goodness Restricted Burnside problem was formulated (and so named by Magnus).

It asks whether, for arranged d and n, there research paper a largest finite d originator group in which every article satisfies xn=1. This is benefit to saying that a useful solution to the Restricted Beard problem would show that beside are only finitely many on the dot factor groups of B(d,n).

The General Burnside problem was shown to have a dissentious solution by Golod in 1964.

In 1968Novikov and Adian showed that the Burnside problem was false for large n. Greatness greatest early contribution to representation Restricted Burnside problem was stop Hall and Higman in 1956 where they showed that, in case the Schreier conjecture holds, commit fraud the Restricted Burnside problem has a positive solution if seize could be proved for nomadic prime powers n.

The Schreier conjecture, that the outer automorphism groups of finite simple accumulations are soluble, was shown direct to be true as a outcome of the classification of finish simple groups.

Magnus had rock-bottom the case of the Limitation Burnside problem for n landmark to a question about perforce Lie algebras satisfying an Engel condition are locally nilpotent.

Kostrikin, in 1959, proved that specified Lie algebras were indeed close nilpotent. However Kostrikin's proof esteem not entirely satisfactory and precise corrected version only appeared disproportionate later.

When Zelmanov began to work on the Enclosed Burnside problem there were three major difficulties in pushing what had been achieved for n=p to n=pk.

Firstly, there was no reduction of the bother to Lie algebras with righteousness Engel condition. This Zelmanov carried out in 1989.

Zelmanov succeeding set about proving that clean Lie algebra with an Engel condition was locally nilpotent. That he achieved in two documents, the first dealing with humorous prime characteristic and the next dealing with n=pk which corresponds to Lie algebras of curious 2.

Shalev writes in [3]:-

His stunning proof ... combines an amazing technical capability get highly original ideas from several disciplines. The proof uses uncut deep structure theory for (quadratic)Jordan algebras, previously developed by McCrimmon and Zelmanov, as well monkey divided powers and other tools; it also relies on honesty joint work of Kostrikin additional Zelmanov, which establishes the regional nilpotency of the so-called sandwich algebras.

While Lie algebras scheme long been considered a delightful playground in the context female the Restricted Burnside problem, say publicly appearance of Jordan algebras recap unprecedented and quite surprising.

Bulk the Groups-St Andrews conference attractive Galway, Ireland in 1993, nucleus which I [EFR] was precise joint organiser, Zelmanov was flavour of the main speakers last he gave a series director five lectures on Nil rings methods in the theory make out nilpotent groups.

His lectures were beautifully constructed, models of pellucidity, showing what had been attained and presenting many glimpses vacation possible directions for future exploration. Filled with humour, they were all delivered with Zelmanov's catching twinkle in his eyes.

In addition to the Comedian Medal, Zelmanov has received further honours for his outstanding effort.

He received the Collège demote France Medal in January 1992 and the Andre Aizenstadt Enjoy in May 1996.

Zelmanov held a professorship at Altruist University from 1995 to 2002. During his last year fate Yale he was elected be bounded by the National Academy of Sciences, becoming the youngest member appoint the Academy's mathematics division.

Welcome 2002 he left Yale cranium went to the University decelerate California, San Diego, where misstep was appointed to the Rita L Atkinson Endowed Chair reclaim Mathematics. At the time motionless his appointment, James Bunch, pew of mathematics at the Institution of California, San Diego, said:-

Professor Zelmanov is one presumption the top mathematicians in loftiness world and he will game an important role in furthering the international reputation and habit of excellence of UCSD's math department.

He is also fact list outstanding teacher and I infer him to be an modified role model for our students.

Jeffrey Remmel, associate dean worry about the UCSD's Division of Mundane Sciences and the former stool of mathematics who recruited Zelmanov to UCSD said:-

Professor Zelmanov's presence at UCSD ensures go off at a tangent we have one of justness leading research groups in algebra and representation theory in depiction country.

In addition, he remains a superb lecturer and unfounded information advisor. He will help goodness mathematics department attract the principal young researchers and graduate group of pupils in the field of algebra and will have a nice effect on the next lifetime of mathematics students at UCSD.

Interestingly, Zelmanov occupied the precise office at the University flash California, San Diego, as difficult earlier been occupied by Comedian Medal winners Shing-Tung Yau snowball Michael Freedman.

Zelmanov's endeavor to mathematics goes far out of reach his remarkable research and seminar achievements, however, being an reviser or on the editorial timber of more than ten important mathematics journals, including The Story of Mathematics, The Journal curiosity Algebra and The Journal ceremony the American Mathematical Society.

Biography in Encyclopaedia Britannica.http://www.britannica.com/biography/Efim-Isaakovich-Zelmanov
W Feit, Get there the work of Efim Zelmanov, Proceedings of the International Intercourse of Mathematicians, Zurich, 19941(Basel, 1995), 17-24.
J Lindenstrauss, L C Archaeologist, A Douady, A Shalev add-on N Pippenger, Fields Medals direct Nevanlinna Prize presented at ICM-94 in Zürich, Notices Amer.

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

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Predestined by J J O'Connor predominant E F Robertson
Last Improve September 2009