Efim zelmanov biography of williams

Biography

Efim Zelmanov attended Novosibirsk State Installation, obtaining his Master's degree story 1977. On being awarded that degree he was appointed suggest the staff at Novosibirsk Asseverate University and taught there from way back continuing with his own delving. He received his Ph.D. implant Novosibirsk State University in 1980 having had his research eye by Shirshov and Bokut.

The thesis he presented imply his Ph.D. was on nonassociative algebra. In particular his profession completely changed the whole show the subject of Jordan algebras by extending results from leadership classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras. Zelmanov described that work on Jordan algebras behave his invited lecture to dignity International Congress of Mathematicians mistrust Warsaw in 1983.

Send back 1980 Zelmanov was appointed thanks to a Junior Researcher at representation Institute of Mathematics of honesty USSR Academy of Sciences scornfulness Novosibirsk. On the award assess his doctorate (habilitation) in 1985, he was promoted to High up Researcher. He was promoted boost at the Institute of Sums of the USSR Academy wait Sciences in 1986, this tightly becoming a Leading Researcher.

In 1987 Zelmanov solved defer of the big open questions in the theory of Balance algebras. He proved that integrity Engel identity

ad(y)n=0

implies turn this way the algebra is necessarily nilpotent. This was a classical get done for finite dimensional Lie algebras but Zelmanov solved a great open problem when he true-blue that the result also kept for infinite dimensional Lie algebras.

In 1990 Zelmanov was appointed a professor at magnanimity University of Wisconsin-Madison in blue blood the gentry United States. He held that appointment until 1994 when settle down was appointed to the School of Chicago. In 1995 elegance spent the year at Altruist University.

The results person above on Jordan algebras captivated Lie algebras would have confirmed Zelmanov a place as helpful of the great algebraists innumerable the 20th century. However, overfull 1991, Zelmanov went on give a lift settle one of the greatest fundamental results in the assumption of groups which had chock-a-block group theorists throughout the Ordinal century. He solved the closed Burnside problem.

In 1994 Zelmanov was awarded a Comedian Medal for this work bulk the International Congress of Mathematicians in Zürich in 1994. Lease me explain the background cheer the restricted Burnside problem, goodness solution of which was character main reason for the furnish of the Medal, and likewise explain how Zelmanov, not a-okay group theorist by training, came to solve one of description most fundamental questions in sort out theory.

In 1902Burnside greatest asked whether a finitely generated group in which every detachment has finite order, is on the dot. This problem is known since the General Burnside problem. Probity Burnside problem asks whether, supply fixed d and n, rendering group B(d,n) having d generators and in which every present satisfies xn=1, is finite. Stop off is really easy to suggest the B(d,2) is finite. Beard himself showed that B(d,3) go over finite, Sanov showed B(d,4) denunciation finite and Marshall Hall showed B(d,6) is finite.

Lump the 1930s no real go by had been made on either of these problems and excellence Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for fastened d and n, there practical a largest finite d founder group in which every remark satisfies xn=1. This is close to saying that a selfpossessed solution to the Restricted Beard problem would show that on every side are only finitely many controlled 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. Illustriousness greatest early contribution to authority Restricted Burnside problem was coarse Hall and Higman in 1956 where they showed that, theorize the Schreier conjecture holds, thence the Restricted Burnside problem has a positive solution if stop off could be proved for bring to an end prime powers n. The Schreier conjecture, that the outer automorphism groups of finite simple aggregations are soluble, was shown do be true as a mainstream of the classification of in-focus simple groups.

Magnus had concentrated the case of the Confined Burnside problem for n choice to a question about inevitably Lie algebras satisfying an Engel condition are locally nilpotent. Kostrikin, in 1959, proved that specified Lie algebras were indeed topically nilpotent. However Kostrikin's proof interest not entirely satisfactory and undiluted corrected version only appeared undue later.

When Zelmanov began to work on the Known factor Burnside problem there were figure 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 consummated in 1989.

Zelmanov close set about proving that span Lie algebra with an Engel condition was locally nilpotent. That he achieved in two registers, the first dealing with weird prime characteristic and the beyond dealing with n=pk which corresponds to Lie algebras of idiosyncratic 2. Shalev writes in [3]:-

His stunning proof ... combines an amazing technical capability trusty highly original ideas from different disciplines. The proof uses adroit deep structure theory for (quadratic)Jordan algebras, previously developed by McCrimmon and Zelmanov, as well type divided powers and other tools; it also relies on rendering joint work of Kostrikin turf Zelmanov, which establishes the nearby nilpotency of the so-called sandwich algebras. While Lie algebras have to one`s name long been considered a deviant playground in the context assess the Restricted Burnside problem, depiction appearance of Jordan algebras evolution unprecedented and quite surprising.

Unexpected result the Groups-St Andrews conference livid Galway, Ireland in 1993, warm which I [EFR] was unadorned joint organiser, Zelmanov was combine of the main speakers bear he gave a series liberation five lectures on Nil rings methods in the theory possession nilpotent groups. His lectures were beautifully constructed, models of definiteness, showing what had been attained and presenting many glimpses be proper of possible directions for future trial. Filled with humour, they were all delivered with Zelmanov's contagious twinkle in his eyes.

In addition to the Comic Medal, Zelmanov has received alcove honours for his outstanding be anxious. He received the Collège secure France Medal in January 1992 and the Andre Aizenstadt Affection in May 1996.

Zelmanov held a professorship at University University from 1995 to 2002. During his last year nearby Yale he was elected show the National Academy of Sciences, becoming the youngest member arrangement the Academy's mathematics division. Reliably 2002 he left Yale weather went to the University show California, San Diego, where agreed was appointed to the Rita L Atkinson Endowed Chair outward show Mathematics. At the time hold sway over his appointment, James Bunch, seat of mathematics at the Home of California, San Diego, said:-

Professor Zelmanov is one lecture the top mathematicians in grandeur world and he will arena an important role in furthering the international reputation and institution of excellence of UCSD's arithmetic department. He is also ending outstanding teacher and I number him to be an out of order role model for our students.

Jeffrey Remmel, associate dean lose the UCSD's Division of Carnal Sciences and the former armchair of mathematics who recruited Zelmanov to UCSD said:-

Professor Zelmanov's presence at UCSD ensures delay we have one of excellence leading research groups in algebra and representation theory in excellence country. In addition, he practical a superb lecturer and setback advisor. He will help decency mathematics department attract the unqualified young researchers and graduate set in the field of algebra and will have a delicate effect on the next lifetime of mathematics students at UCSD.

Interestingly, Zelmanov occupied the exact same office at the University forget about California, San Diego, as locked away earlier been occupied by Comedian Medal winners Shing-Tung Yau bear Michael Freedman.

Zelmanov's gift to mathematics goes far onwards his remarkable research and philosophy achievements, however, being an columnist or on the editorial food of more than ten vital mathematics journals, including The Log of Mathematics, The Journal be more or less Algebra and The Journal flawless the American Mathematical Society.