Algebraic, Geometric, and Asymptotic Properties of Branch Groups

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DMS-0308985Grigorchuk, RostislavAbstractTitle: Algebraic, Geometric, and Asymptotic Properties of Branch GroupsThis project will focus on properties of branch groups.Branch groups are abstract mathematical objects that play an important role in solving major problems on the intersection of different parts of mathematics, such as Algebra, Number Theory, Analysis, Geometry and Probability Theory. The proposer will study these groups and their relation to a number of important and long-standing problems in modern mathematics, such as the Burnside and Milnor Problems in Group Theory, Atiyah Conjecture in Topology, Kadison-Kaplansky Conjecture in Functional Analysis. The results obtained in this project have useful applications in Cybernetics and Computer Science, as well as Cryptography and Theory of Algorithms.Combinatorial Group Theory is one of the most rapidly developing parts of modern mathematics. Starting with works by Poincare and Klein this area received contribution from some of the most prominent mathematicians of the twentieth century, such as von Neumann, Milnor and others. In recent decades Combinatorial Group Theory became an object of especially intensive research due to newly found connections with Statistical Physics, Quantum Chaos Theory and Computer Science. New promising directions of mathematical research have been developped, one of them being the theory of branch groups. This theory has strong connections to several mainstream open problems in modern mathematics. Tremendous progress acheived in recent years indicates that the exploration of these connections leads to most fruitful ideas and results.