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Colloquium October 29

Date: Friday October 29

Time: 16:00 to 17:00

Place: 455 McBryde (Commons Room)

Speaker: Rostislav Grigorchuk of Texas A&M University

Title: Algebraic and spectral properties of groups generated by finite automata

Abstract

Automata groups were defined 40 years ago but started to play important role only two decades later after discovery that they can give a solution to famous problems such as Burnside problem on torsion groups or Milnor problem on group of intermediate growth. The development of the theory of automata groups during last two decades showed that the number of difficult problems to which methods and ideas of groups generated by finite automata can be applied is much larger and that they can be used in many areas of mathematics such as algebra, analysis, geometry, probability, dynamics, computer science, mathematical logic and other. Automata groups were used to solve several difficult problems in analysis around the notion of amenability introduced by von Neumann in 1929 (as a result of study of algebraical roots of Banach-Tarskii paradox), in Riemannian geometry (around Atiyah Problem on L2-invariants), in the theory of profinite groups (Zelmanov Conjecture on groups of finite width), in geometric group theory (Gromov problem on groups of uniformly exponential growth). There are indications that they can be useful to attack the Kaplanskii Conjecture on idempotents and the Kaplanskii Conjecture on the Jacobson radical, Mazur-Fontain Conjecture in Number theory and other problems of mathematics. We will give a survey of results and methods.


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