Topology, Algebra and Markov Chains

建议：DMS-0071428 PI：Kenneth BrownInstitution：Cornell摘要：该提案涵盖了与拓扑学、代数学、组合学和概率相关的广泛主题。 具体来说，首席研究员建议研究：有限群陪集的偏序集;与超平面排列，Coxeter群和建筑物相关的马尔可夫链，包括与人口遗传学中的Moran模型相关的链; Artin群在无穷远处的拓扑;某些无限群上随机游动的谱半径;和有限群的生成集的图。这项工作揭示了不同数学领域之间意想不到的联系，从而照亮了这两个领域。 举例来说：（a）抽象组合对象上的随机游走揭示了群体遗传学中的一个模型;（B）几何学和代数学导致了关于马尔可夫链的新结果，马尔可夫链用于科学和应用数学的许多领域的模拟;（c）关于群的概率问题，群是用于对称性研究的代数对象，导致了研究这些群的新拓扑工具。

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.