Topology, Algebra and Markov Chains

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Proposal: DMS-0071428PI: Kenneth BrownInstitution: CornellAbstract:This proposal covers a wide range of topics relating topology, algebra, combinatorics, and probability. Specifically, the principal investigator proposes to study: The poset of cosets of a finite group; Markov chains associated with hyperplane arrangements, Coxeter groups, and buildings, including a chain that is related to the Moranmodel in population genetics; the topology at infinity of Artin groups; the spectral radius of random walks on certain infinite groups; and the graph of generating sets of a finite group.This work reveals unexpected connections between different fields of mathematics, thereby illuminating both fields. For example: (a) a random walk on an abstract combinatorial object sheds light on a model in population genetics; (b) geometry and algebra have led to new results about Markov chains, which are used for simulations in many areas of science and applied mathematics; and (c) a probabilistic question about groups, which are algebraic objects used in the study of symmetry, leads to a new topological tool for investigating these groups.