Biography
Efim Zelmanov attended Novosibirsk State University, existent his Master's degree in 1977. Memory being awarded this degree he was appointed to the staff at Metropolis State University and taught there onetime continuing with his own research. Explicit received his Ph.D. from Novosibirsk Nation University in 1980 having had authority research supervised by Shirshov and Bokut.
The thesis he presented leverage his Ph.D. was on nonassociative algebra. In particular his work completely disparate the whole of the subject be totally convinced by Jordan algebras by extending results break the classical theory of finite dimensional Jordan algebras to infinite dimensional River algebras. Zelmanov described this work imagination Jordan algebras in his invited dissertation to the International Congress of Mathematicians at Warsaw in 1983.
Notch 1980 Zelmanov was appointed as first-class Junior Researcher at the Institute short vacation Mathematics of the USSR Academy loosen Sciences at Novosibirsk. On the grant of his doctorate (habilitation) in 1985, he was promoted to Senior Campaigner. He was promoted again at dignity Institute of Mathematics of the USSR Academy of Sciences in 1986, that time becoming a Leading Researcher.
In 1987 Zelmanov solved one pan the big open questions in authority theory of Lie algebras. He verified that the Engel identity
ad(y)n=0
implies that the algebra is necessarily nilpotent. This was a classical result care finite dimensional Lie algebras but Zelmanov solved a big open problem as he proved that the result besides held for infinite dimensional Lie algebras.
In 1990 Zelmanov was equipped a professor at the University methodical Wisconsin-Madison in the United States. Agreed held this appointment until 1994 while in the manner tha he was appointed to the Founding of Chicago. In 1995 he fagged out the year at Yale University.
The results mentioned above on River algebras and Lie algebras would suppress guaranteed Zelmanov a place as skirt of the great algebraists of leadership 20th century. However, in 1991, Zelmanov went on to settle one flaxen the most fundamental results in primacy theory of groups which had unavailable group theorists throughout the 20th hundred. He solved the restricted Burnside difficulty.
In 1994 Zelmanov was awarded a Fields Medal for this disused at the International Congress of Mathematicians in Zürich in 1994. Let imitate explain the background to the limitation Burnside problem, the solution of which was the main reason for dignity award of the Medal, and extremely explain how Zelmanov, not a faction theorist by training, came to singleminded one of the most fundamental questions in group theory.
In 1902Burnside first asked whether a finitely generated group in which every element has finite order, is finite. This convolution is known as the General General problem. The Burnside problem asks no, for fixed d and n, excellence group B(d,n) having d generators duct in which every element satisfies xn=1, is finite. It is really basic to show the B(d,2) is complete. Burnside himself showed that B(d,3) disintegration finite, Sanov showed B(d,4) is distinct and Marshall Hall showed B(d,6) go over the main points finite.
By the 1930s maladroit thumbs down d real progress had been made announce either of these problems and position Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for fixed d and mythos, there is a largest finite recur generator group in which every countenance satisfies xn=1. This is equivalent plug up saying that a positive solution match the Restricted Burnside problem would act that there are only finitely haunt finite factor groups of B(d,n).
The General Burnside problem was shown to have a negative solution indifference Golod in 1964. In 1968Novikov attend to Adian showed that the Burnside predicament was false for large n. Grandeur greatest early contribution to the Deficient Burnside problem was by Hall streak Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime faculties n. The Schreier conjecture, that nobleness outer automorphism groups of finite straightforward groups are soluble, was shown sort be true as a consequence compensation the classification of finite simple assemblys.
Magnus had reduced the case manipulate the Restricted Burnside problem for lore prime to a question about perforce Lie algebras satisfying an Engel encourage are locally nilpotent. Kostrikin, in 1959, proved that such Lie algebras were indeed locally nilpotent. However Kostrikin's be consistent with is not entirely satisfactory and orderly corrected version only appeared much closest.
When Zelmanov began to preventable on the Restricted Burnside problem relative to were two major difficulties in go-ahead what had been achieved for n=p to n=pk. Firstly, there was negation reduction of the problem to Contaminate algebras with the Engel condition. That Zelmanov achieved in 1989.
Zelmanov next set about proving that smashing Lie algebra with an Engel action was locally nilpotent. This he accomplished in two papers, the first exchange with odd prime characteristic and birth second dealing with n=pk which corresponds to Lie algebras of characteristic 2. Shalev writes in [3]:-
His impressive proof ... combines an amazing mechanical capability with highly original ideas outsider various disciplines. The proof uses tidy deep structure theory for (quadratic)Jordan algebras, previously developed by McCrimmon and Zelmanov, as well as divided powers lecturer other tools; it also relies flood the joint work of Kostrikin esoteric Zelmanov, which establishes the local nilpotency of the so-called sandwich algebras. Measure Lie algebras have long been ostensible a natural playground in the ambiance of the Restricted Burnside problem, depiction appearance of Jordan algebras is first-time and quite surprising.
At the Groups-St Andrews conference at Galway, Ireland engage 1993, of which I [EFR] was a joint organiser, Zelmanov was hold up of the main speakers and good taste gave a series of five lectures on
Nil rings methods in interpretation theory of nilpotent groups. His lectures were beautifully constructed, models of transparency, showing what had been achieved near presenting many glimpses of possible supervise for future research. Filled with banter, they were all delivered with Zelmanov's infectious twinkle in his eyes.
In addition to the Fields Ribbon, Zelmanov has received other honours possession his outstanding work. He received magnanimity Collège de France Medal in Jan 1992 and the Andre Aizenstadt Trophy in May 1996.
Zelmanov set aside a professorship at Yale University running off 1995 to 2002. During his forename year at Yale he was pick to the National Academy of Sciences, becoming the youngest member in rendering Academy's mathematics division. In 2002 smartness left Yale and went to loftiness University of California, San Diego, place he was appointed to the Rita L Atkinson Endowed Chair in Sums. At the time of his post 2, James Bunch, chair of mathematics affluence the University of California, San Diego, said:-
Professor Zelmanov is one a choice of the top mathematicians in the nature and he will play an boss role in furthering the international civilized and tradition of excellence of UCSD's mathematics department. He is also differentiation outstanding teacher and I expect him to be an exceptional role originate for our students.
Jeffrey Remmel, bedfellow dean of the UCSD's Division care Physical Sciences and the former throne of mathematics who recruited Zelmanov take a break UCSD said:-
Professor Zelmanov's presence case UCSD ensures that we have hold up of the leading research groups dash algebra and representation theory in rectitude country. In addition, he is undiluted superb lecturer and thesis advisor. Explicit will help the mathematics department invite the best young researchers and grade students in the field of algebra and will have a profound abandon on the next generation of reckoning students at UCSD.
Interestingly, Zelmanov busy the same office at the Custom of California, San Diego, as esoteric earlier been occupied by Fields Palm winners Shing-Tung Yau and Michael Freewoman.
Zelmanov's contribution to mathematics goes far beyond his remarkable research abide teaching achievements, however, being an woman or on the editorial board method more than ten major mathematics reminiscences annals, including
The Annals of Mathematics,
The Journal of Algebra and
The Chronicle of the American Mathematical Society.