Amenabilty, soficity, dynamics, and operator algebras
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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.