Harald Bohr Lecture by Efim Zelmanov

Efim Zelmanov writes in his abstract: ”We will discuss open problems related to residually finite groups and infinite dimensional algebras. As is typical in Combinatorial Group Theory the problems will be inspired by (i) The Burnside Problem, (ii) growth, (iii) presentations by generators and relations.”

Efim Zelmanov

Professor Efim Zelmanov is one of today’s most distinguished mathematicians. He is the recipient of a 1994 Fields Medal, in recognition of his groundbreaking work in the field of abstract algebra, including group theory, and specifically for his solution of the Restricted Burnside Problem. In addition to the Fields Medal, Professor Zelmanov has received other honours for his outstanding work, including the Collège de France Medal (1992) and the Andre Aizenstadt Prize (1996).

Efim Zelmanov studied Mathematics at the State University in Novosibirsk, Russia. Upon receiving his Ph.D. in 1980, he was appointed Researcher at the Institute of Mathematics of the USSR Academy of Sciences at Novosibirsk. In 1990 he became Professor at the University of Wisconsin-Madison. Following appointments at the University of Chicago (1994-1995) and Yale University (1995-2002), he is currently holding the Rita L. Atkinson Chair in Mathematics at the University of California, San Diego.

Professor Zelmanov is a member of the US National Academy of Science, the American Academy of Arts and Sciences, the Korean Academy of Sciences, the Royal Academy of Spain, and the Academy of Sciences of Brazil.

Here are excerpts (translated from German) from the Laudatio delivered by Walter Feit (Yale University) at the Fields Medal Award ceremony, International Congress of Mathematicians, Zurich, 1994:

"Efim Zelmanov received the Fields Medal for his brilliant solution to the long-time open Restricted Burnside Problem. It is a problem deep-rooted in group theory, the basis for the mathematical study of symmetries. What is asked for is a bound for the number of symmetries of an object, when each symmetry has bounded order."

”Prior to the solution of the Restricted Burnside Problem, Zelmanov had already made important contributions to the theory of Lie algebras and to that of Jordan algebras; these theories have their origins in geometry, respectively in quantum mechanics. Some of his results that were achieved there were of crucial importance for his group theoretical work. In this way, the unity of mathematics is once again documented and it shows how much seemingly far apart areas are connected and influence each other."

Jonathan Alperin (University of Chicago) called Zelmanov's solution of the Restricted Burnside Problem "a tremendous breakthrough and a real triumph.”

"His proof is awesome," Alperin said, "and at a level completely ahead of any previous work in algebra. It is notable for its sheer power, but also for the volcano of new techniques he developed to bring it to completion. His single effort has completely changed and advanced the whole field of algebra and given us a whole new field on which to operate.”

Magdalena Musat