Rostislav Grigorchuk of Texas A&M University will be awarded the 2015 AMS Leroy P. Steele Prize for Seminal Contribution to Research at the Joint Mathematics Meetings in January in San Antonio, Texas. Grigorchuk is honored for his influential paper 'Degrees of growth of finitely generated groups and the theory of invariant means,' which appeared in Russian in 1984 in Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya and in English translation a year later.
This paper gave the first account of Grigorchuk's remarkable discovery of a new class of groups that established counterintuitive results and opened new vistas. The paper stands as a landmark in the development of the now-burgeoning area of geometric group theory.
In mathematics, a group is a set whose elements can be multiplied together, or composed. Any group can be thought of as a group of symmetries, so that multiplication is just the composition of symmetries. A finitely generated group is one that, though usually infinite, has a finite set of elements, called generators, whose iterated products give all group elements.
One can ask how efficiently a finite set of generators generates an infinite group. To this end one looks at how quickly the number of elements obtained by iterated products grows with the number of iterations.
In 1968, John Milnor conjectured that this growth would be either exponential or polynomial, meaning that the number of elements obtained by n-fold products would be either an exponential or a polynomial function of n. Grigorchuk, in his prize-winning paper, discovered previously unknown groups that exhibit "intermediate" growth rates that fall between the two rates in Milnor's conjecture. This means that the number of elements obtained by n-fold products grows faster than any polynomial function of n, but is still sub-exponential. In addition to establishing this very surprising result, the groups Grigorchuk discovered, which have since been named "Grigorchuk groups", have many other amazing properties.
"Grigorchuk's work not only gave solutions to old standing problems but also discovered new exciting classes of groups," the prize citation says. "They found applications in the theory of fractals, holomorphic dynamics, spectral theory of groups and graphs, and theory of finite automata... The work of Grigorchuk has influenced several generations of researchers in group theory. It would be impossible to imagine the modern group theory without Grigorchuk's work."
Presented annually, the AMS Steele Prize is one of the highest distinctions in mathematics. The prize will be awarded at the Joint Mathematics Meetings, Sunday, January 11, 2015 at 4:25 PM, at the Henry B. Gonzalez Convention Center in San Antonio, Texas.
Find out more about AMS prizes and awards at http://ams.org/prizes-awards/prizes.
Founded in 1888 to further mathematical research and scholarship, today the nearly 30,000 member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.