CAREER: Amenable and recurrent actions of finitely generated groups

翻译后摘要（Juschenko，1352173）：“顺从性”的主题基本上开始于20世纪初与勒贝格。他问是否他的积分性质是真正的基本和遵循更熟悉的积分公理。 顺从群的类是由冯·诺依曼在1929年引入并研究的，他解释了为什么悖论只出现在大于或等于3的维度中。在1940年代，顺从性理论转移到功能分析领域。 目前顺从性理论出现在许多数学领域，最显着的是在算子代数，泛函分析，遍历理论，概率论和调和分析。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体是顺从的。PI计划使用Schreier图上的随机游走以更概念化的方式开发这种技术。有几类群体可以潜在地应用该技术。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更深入地理解群体行动的分析性质，并将其与其他领域联系起来。这项建议的主要影响之一是通过其教育目标，特别是组织几个暑期学校和讲习班。 这些学校的目标是研究生和其他年轻的研究人员，专注于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.