Amenabilty, soficity, dynamics, and operator algebras

泛函分析为研究无限维结构提供了一种强有力的数学语言。这种语言的抽象性质允许广泛的应用，其各种设置包括量子力学，概率论和系统的研究，物理或其他，因为它们随着时间的推移而演变。该项目将集中在这三个设置的最后几个方面和广义形式，它将主要围绕与熵有关的现象。有限或有限维近似长期以来一直在动力学和算子代数中发挥着重要作用，并通过顺从性和soficity及其代数类似物的概念以其最基本和最有影响力的形式表现出来。在C*-代数域中，顺从性受到各种各样的改进，如有限分解秩和有限核维数，它们表达了艾略特程序中K理论分类定理所必需的拓扑正则性。本项目旨在拓宽我们对拓扑动力学中这些规律性性质和现象之间关系的理解，特别是平均维数和Rokhlin型维数不变量，并在迄今为止仍然超出分类范围的各种例子上测试这种理解，例如分支群的作用。组合独立性将作为分析动力学中的混合性和矛盾性及其与拟对角性和维数行为的关系的工具，并将在熵和轨道等价的研究中应用于sofic群的作用。

The subject of functional analysis provides a powerful mathematical language for the investigation of infinite-dimensional structures. The abstract nature of this language allows for a wide spectrum of applications, whose various settings include quantum mechanics, the theory of probability, and the study of systems, physical or otherwise, as they evolve in time. The project will concentrate on several aspects and generalized forms of the last of these three settings, and it will largely revolve around phenomena related to entropy.Finite or finite-dimensional approximation has long played a fundamental role in both dynamics and operator algebras and is manifest in its most basic and influential forms through the notions of amenability and soficity and their algebraic analogues. In the domain of C*-algebras, amenability has been subject to various refinements like finite decomposition rank and finite nuclear dimension that express the kind of topological regularity necessary for K-theoretic classification theorems within the Elliott program. This project aims to broaden our understanding of the relationships between these regularity properties and phenomena in topological dynamics, in particular mean dimension and Rokhlin-type dimensional invariants, and to test this understanding on a variety of examples that have so far remained beyond the scope of classification, such as actions of branch groups. Combinatorial independence will be brought into this picture as a tool for analyzing mixing and paradoxicality in dynamics and their relation to quasidiagonality and dimensional behavior, and it will be also applied to actions of sofic groups in the study of entropy and orbit equivalence.