CAREER: Amenable and recurrent actions of finitely generated groups

摘要(Juschenko，1352173)：“顺从”的主题基本上始于1900年代的勒贝格。他问他的积分的性质是否真的是基本的，并遵循更熟悉的积分公理。Von Neumann在1929年引入并研究了顺从群这一类，他解释了为什么悖论只出现在大于或等于3的维度上。在20世纪40年代，顺从性理论转向了功能分析领域。目前，顺从性理论出现在许多数学领域，特别是在算子代数、泛函分析、遍历理论、概率论和调和分析中。该项目的核心部分是基于PI最近使用的技术来证明某些群是顺从性的。PI计划使用Schreier图上的随机游动，以更概念化的方式开发这项技术。这项技术可以潜在地应用于几类群体。它们包括Thompson群F，区间交换变换群，交换群的拓扑全群。该项目的预期影响是加深对群体行动的分析属性的理解，并将它们与其他领域联系起来。这项建议的核心影响之一是通过其教育目标，特别是在组织几个暑期学校和讲习班方面。这些学校面向研究生和其他年轻的研究人员，重点关注群论和算子代数的最新发展。

Abstract (Juschenko, 1352173): The subject of "amenability" essentially begins in the 1900s with Lebesgue. He asked whether the properties of his integral are really fundamental and follow from more familiar integral axioms. The class of amenable groups was introduced and studied by von Neumann in 1929, and he explained why a paradox appeared only in dimensions greater or equal to three. In the 1940s the amenability theory shifted into the field of functional analysis. Currently amenability theory appears in many fields of mathematics, most notably in operator algebras, functional analysis, ergodic theory, probability theory, and harmonic analysis.The core part of the project is based on the recent technique used by PI to prove that certain groups are amenable. PI plans to develop this technique in a more conceptual way using random walks on Schreier graphs. There are several classes of groups to which the technique can potentially be applied. They include Thompson group F, interval exchange transformation group, topological full group of abelian groups. The expected impact of the project is to give deeper understanding of analytic properties of the group actions and relate them to other fields. One of the central impact of this proposal is through its educational goals, in particular, in organizing of several summer schools and workshops. The schools are aimed toward graduate students and other young researchers, focused on the recent developments in group theory and operator algebras.