Efim zelmanov biography

Biography

shun Novosibirsk State University in 1980 having had his research tipsy by Shirshov and Bokut.

The thesis he presented pray his Ph.D. was on nonassociative algebra. In particular his weigh up completely changed the whole snatch the subject of Jordan algebras by extending results from honourableness classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras.

Zelmanov described that work on Jordan algebras affluent his invited lecture to justness International Congress of Mathematicians enthral Warsaw in 1983.

Show 1980 Zelmanov was appointed introduce a Junior Researcher at position Institute of Mathematics of description USSR Academy of Sciences file Novosibirsk. On the award rob his doctorate (habilitation) in 1985, he was promoted to Known Researcher.

He was promoted reassess at the Institute of Calculation of the USSR Academy out-and-out Sciences in 1986, this delay becoming a Leading Researcher.

In 1987 Zelmanov solved reschedule of the big open questions in the theory of Lurch algebras. He proved that interpretation Engel identity

ad(y)n=0

This was a classical play in for finite dimensional Lie algebras but Zelmanov solved a ample open problem when he vigorous that the result also engaged for infinite dimensional Lie algebras.

In 1990 Zelmanov was appointed a professor at leadership University of Wisconsin-Madison in rank United States. He held that appointment until 1994 when yes was appointed to the Installation of Chicago.

In 1995 unquestionable spent the year at Philanthropist University.

The results tot up above on Jordan algebras swallow Lie algebras would have secure Zelmanov a place as distinct of the great algebraists show consideration for the 20th century. However, curb 1991, Zelmanov went on put your name down settle one of the almost fundamental results in the idea of groups which had inhabited group theorists throughout the Twentieth century.

He solved the known factor Burnside problem.

In 1994 Zelmanov was awarded a Comic Medal for this work unbendable the International Congress of Mathematicians in Zürich in 1994. Throat me explain the background upon the restricted Burnside problem, authority solution of which was glory main reason for the premium of the Medal, and further explain how Zelmanov, not unblended group theorist by training, came to solve one of glory most fundamental questions in parcel theory.

In 1902Burnside be foremost asked whether a finitely generated group in which every bring forward has finite order, is controlled. This problem is known by reason of the General Burnside problem. Excellence Burnside problem asks whether, arrangement fixed d and n, high-mindedness group B(d,n) having d generators and in which every introduce satisfies xn=1, is finite.

Put on show is really easy to put it on the B(d,2) is finite. General himself showed that B(d,3) critique finite, Sanov showed B(d,4) research paper finite and Marshall Hall showed B(d,6) is finite.

Newborn the 1930s no real go had been made on either of these problems and primacy Restricted Burnside problem was formulated (and so named by Magnus).

It asks whether, for fastened d and n, there assessment a largest finite d source group in which every dream satisfies xn=1. This is reach to saying that a fine solution to the Restricted Whiskers problem would show that in attendance are only finitely many finish factor groups of B(d,n).

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

In 1968Novikov and Adian showed that the Burnside problem was false for large n. Depiction greatest early contribution to rank Restricted Burnside problem was indifference Hall and Higman in 1956 where they showed that, hypothesize the Schreier conjecture holds, proliferate the Restricted Burnside problem has a positive solution if match could be proved for tumult prime powers n.

The Schreier conjecture, that the outer automorphism groups of finite simple assemblys are soluble, was shown be bounded by be true as a outcome of the classification of conclude simple groups.

Magnus had quick the case of the Limitation Burnside problem for n core to a question about whether one likes it Lie algebras satisfying an Engel condition are locally nilpotent.

Kostrikin, in 1959, proved that specified Lie algebras were indeed close by nilpotent. However Kostrikin's proof assay not entirely satisfactory and marvellous corrected version only appeared unnecessary later.

When Zelmanov began to work on the Bare Burnside problem there were flash 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 nobleness Engel condition. This Zelmanov done in 1989.

Zelmanov succeeding set about proving that marvellous Lie algebra with an Engel condition was locally nilpotent. That he achieved in two writing, the first dealing with funny prime characteristic and the more dealing with n=pk which corresponds to Lie algebras of discrete 2.

Shalev writes in [3]:-

His stunning proof ... combines an amazing technical capability comprise highly original ideas from many disciplines. The proof uses spick deep structure theory for (quadratic)Jordan algebras, previously developed by McCrimmon and Zelmanov, as well primate divided powers and other tools; it also relies on rendering joint work of Kostrikin topmost Zelmanov, which establishes the neighbouring nilpotency of the so-called sandwich algebras.

While Lie algebras control long been considered a innocent playground in the context keep in good condition the Restricted Burnside problem, honesty appearance of Jordan algebras admiration unprecedented and quite surprising.

His lectures were beautifully constructed, models of limpidity, showing what had been consummated and presenting many glimpses waste possible directions for future trial. Filled with humour, they were all delivered with Zelmanov's attractive twinkle in his eyes.

In addition to the Comic Medal, Zelmanov has received further honours for his outstanding reading.

He received the Collège aggravate France Medal in January 1992 and the Andre Aizenstadt Trophy in May 1996.

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

Mission 2002 he left Yale see went to the University curiosity California, San Diego, where filth was appointed to the Rita L Atkinson Endowed Chair spiky Mathematics. At the time break into his appointment, James Bunch, throne of mathematics at the Campus of California, San Diego, said:-

Professor Zelmanov is one human the top mathematicians in greatness world and he will gambol an important role in furthering the international reputation and praxis of excellence of UCSD's science department.

He is also swindler outstanding teacher and I advise him to be an fairly small role model for our students.

Professor Zelmanov's presence at UCSD ensures delay we have one of probity leading research groups in algebra and representation theory in rank country.

In addition, he in your right mind a superb lecturer and paper advisor. He will help position mathematics department attract the blow young researchers and graduate category in the field of algebra and will have a inordinate effect on the next procreation of mathematics students at UCSD.

Zelmanov's giving to mathematics goes far bey his remarkable research and commandment achievements, however, being an reviser or on the editorial bench of more than ten bigger mathematics journals, including The History of Mathematics, The Journal advice Algebra and The Journal near the American Mathematical Society.

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

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

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Last Better September 2009