CAREER: Amenable and recurrent actions of finitely generated groups

摘要（Juschenko, 1352173）：“顺从性”的主题基本上始于20世纪的勒贝格。他问他的积分的性质是否真的是基本的，是否遵循更熟悉的积分公理。1929年，冯·诺伊曼引入并研究了可服从群这一概念，他解释了为什么只有在大于或等于3的维度上才会出现悖论。20世纪40年代，适应性理论转移到功能分析领域。目前，适应性理论出现在数学的许多领域，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体是可服从的。PI计划在Schreier图上使用随机漫步，以更概念化的方式开发这项技术。该技术可以应用于几类群体。它们包括汤姆逊群F、区间交换变换群、阿贝尔群的拓扑满群。该项目的预期影响是对群体行为的分析特性有更深的理解，并将它们与其他领域联系起来。该提案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.