Research in Combinatorial Algebra

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ZELMANOV, 97-04132 We focus on a number of longstanding open problems in algebra and topology that are related to the Burnside Problem. A secondary topic is the study of narrow groups and Lie algebras, in particular, we consider the classification of the so called superconformal algebras and of pro-unipotent groups of maximal class. The area of proposed research is Combinatorial Algebra. Combinatorial Group Theory appeared about a hundred years ago as a study of certain infinite groups that arise in Topology. Throughout these years it has more or less focused on the following three mainstream problems : (i) groups presented by generators and relators, (ii) growth, (iii) the Burnside Problem. Starting with 1940s similar problems have been considered and intensively studied for infinite - dimensional associative and Lie algebras. This extension of scope of research was motivated (a) by the intrinsic needs of Algebra and in particular of Combinatorial Group Theory, (b) by applications to Functional Analysis, Representation Theory, Number Theory and (increasingly in recent years) by Mathematical Physics.