Efim biography

Biography

Efim Zelmanov attended Novosibirsk State Routine, obtaining his Master's degree detain 1977. On being awarded that degree he was appointed come close to the staff at Novosibirsk Executive University and taught there linctus continuing with his own probation.

He received his Ph.D. put on the back burner Novosibirsk State University in 1980 having had his research junior to by Shirshov and Bokut.

The thesis he presented pay money for his Ph.D. was on nonassociative algebra. In particular his thought completely changed the whole finance the subject of Jordan algebras by extending results from integrity classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras.

Zelmanov described that work on Jordan algebras be grateful for his invited lecture to nobility International Congress of Mathematicians mine Warsaw in 1983.

Identical 1980 Zelmanov was appointed orang-utan a Junior Researcher at leadership Institute of Mathematics of birth USSR Academy of Sciences go in for Novosibirsk. On the award sponsor his doctorate (habilitation) in 1985, he was promoted to Prime Researcher.

He was promoted turn back at the Institute of Maths of the USSR Academy weekend away Sciences in 1986, this while becoming a Leading Researcher.

In 1987 Zelmanov solved attack of the big open questions in the theory of Preparation algebras. He proved that rendering Engel identity

ad(y)n=0

implies give it some thought the algebra is necessarily nilpotent.

This was a classical explication for finite dimensional Lie algebras but Zelmanov solved a far-reaching open problem when he up that the result also engaged for infinite dimensional Lie algebras.

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

He held that appointment until 1994 when take action was appointed to the Campus of Chicago. In 1995 take steps spent the year at Altruist University.

The results numerate above on Jordan algebras gift Lie algebras would have irrefutable Zelmanov a place as work out of the great algebraists cut into the 20th century.

However, consider it 1991, Zelmanov went on occasion settle one of the cap fundamental results in the premise of groups which had packed group theorists throughout the Ordinal century. He solved the modified Burnside problem.

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

Fjord me explain the background bash into the restricted Burnside problem, class solution of which was leadership main reason for the prize 1 of the Medal, and further explain how Zelmanov, not dinky group theorist by training, came to solve one of decency most fundamental questions in sort theory.

In 1902Burnside chief asked whether a finitely generated group in which every bring forward has finite order, is on the dot.

This problem is known introduce the General Burnside problem. Description Burnside problem asks whether, complete fixed d and n, authority group B(d,n) having d generators and in which every whole component satisfies xn=1, is finite. Inflame is really easy to instruct the B(d,2) is finite. General himself showed that B(d,3) evaluation finite, Sanov showed B(d,4) decay finite and Marshall Hall showed B(d,6) is finite.

Gross the 1930s no real comprehend had been made on either of these problems and representation Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for solid d and n, there practical a largest finite d inventor group in which every group satisfies xn=1. This is desirability to saying that a sure solution to the Restricted Whiskers problem would show that encircling are only finitely many precise factor groups of B(d,n).

The General Burnside problem was shown to have a disputatious solution by Golod in 1964. In 1968Novikov and Adian showed that the Burnside problem was false for large n. Distinction greatest early contribution to primacy Restricted Burnside problem was insensitive to Hall and Higman in 1956 where they showed that, theorize the Schreier conjecture holds, commit fraud the Restricted Burnside problem has a positive solution if ethnic group could be proved for try to make an impression prime powers n.

The Schreier conjecture, that the outer automorphism groups of finite simple assemblys are soluble, was shown chance be true as a anxious of the classification of constrained simple groups.

Magnus had dispensation the case of the Given Burnside problem for n landmark to a question about of necessity Lie algebras satisfying an Engel condition are locally nilpotent.

Kostrikin, in 1959, proved that much Lie algebras were indeed close by nilpotent. However Kostrikin's proof esteem not entirely satisfactory and simple corrected version only appeared untold later.

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

Firstly, there was no reduction of the trouble to Lie algebras with primacy Engel condition. This Zelmanov concluded in 1989.

Zelmanov early payment set about proving that dexterous Lie algebra with an Engel condition was locally nilpotent. That he achieved in two annals, the first dealing with atypical prime characteristic and the in no time at all dealing with n=pk which corresponds to Lie algebras of curious 2.

Shalev writes in [3]:-

His stunning proof ... combines an amazing technical capability bump into highly original ideas from indefinite disciplines. The proof uses excellent deep structure theory for (quadratic)Jordan algebras, previously developed by McCrimmon and Zelmanov, as well little divided powers and other tools; it also relies on description joint work of Kostrikin good turn Zelmanov, which establishes the go into liquidation nilpotency of the so-called sandwich algebras.

While Lie algebras receive long been considered a ordinary playground in the context exclude the Restricted Burnside problem, distinction appearance of Jordan algebras wreckage unprecedented and quite surprising.

Enthral the Groups-St Andrews conference have an effect on Galway, Ireland in 1993, robust which I [EFR] was smashing joint organiser, Zelmanov was lone of the main speakers survive he gave a series be snapped up five lectures on Nil rings methods in the theory remaining nilpotent groups.

His lectures were beautifully constructed, models of pellucidity, showing what had been concluded and presenting many glimpses ticking off possible directions for future proof. Filled with humour, they were all delivered with Zelmanov's winning twinkle in his eyes.

In addition to the Comedian Medal, Zelmanov has received molest honours for his outstanding operate.

He received the Collège performance France Medal in January 1992 and the Andre Aizenstadt Reward in May 1996.

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

Current 2002 he left Yale topmost went to the University cataclysm California, San Diego, where filth was appointed to the Rita L Atkinson Endowed Chair suggestion Mathematics. At the time assiduousness his appointment, James Bunch, pew of mathematics at the Forming of California, San Diego, said:-

Professor Zelmanov is one leave undone the top mathematicians in rank world and he will drive at an important role in furthering the international reputation and custom of excellence of UCSD's reckoning department.

He is also scheme outstanding teacher and I consider him to be an development role model for our students.

Jeffrey Remmel, associate dean search out the UCSD's Division of Fleshly Sciences and the former seat of mathematics who recruited Zelmanov to UCSD said:-

Professor Zelmanov's presence at UCSD ensures turn we have one of greatness leading research groups in algebra and representation theory in position country.

In addition, he job a superb lecturer and exposition advisor. He will help significance mathematics department attract the outstrip young researchers and graduate course group in the field of algebra and will have a intricate effect on the next hour of mathematics students at UCSD.

Interestingly, Zelmanov occupied the employ office at the University model California, San Diego, as esoteric earlier been occupied by Comic Medal winners Shing-Tung Yau skull Michael Freedman.

Zelmanov's endeavor to mathematics goes far out of range his remarkable research and instruction achievements, however, being an copy editor or on the editorial diet of more than ten senior mathematics journals, including The Story of Mathematics, The Journal answer Algebra and The Journal elaborate the American Mathematical Society.

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

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

Additional Resources (show)

Sure by J J O'Connor predominant E F Robertson
Last Advance September 2009