Topology and Measure in Dynamics and Operator Algebras

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Three of the basic ingredients in the structural foundations of modern analysis and its connections with theoretical physics are the concepts of measure, topology, and group. The first of these deals with the notions of volume and size, the second with proximity and convergence, and the third with symmetry and the idea of displacement in space or time. When combined together they form the subject of dynamical systems, which in its classical origins models the time evolution of physical systems but is nowadays applicable to a wide range of phenomena involving transformations from one state to another. The project will pursue novel relationships between measure and topology that have recently begun to emerge within this dynamical framework and are tightly linked to the related field of operator algebras. The goal is to use this perspective to develop new ways of understanding how the particular symmetries of a given dynamical system may condition different types of asymptotic behavior, from the deterministic to the chaotic.While the topological and measure-theoretic perspectives have long between fruitfully intertwined in the theory of operator algebras, in the last several years this symbiosis has been reinvigorated through not only the elaboration of surprisingly far-reaching analogies but also the discovery of new kinds of applications of von Neumann algebra techniques to C*-algebras with structurally profound consequences. The central aim of the project is to reimagine these analogies and techniques in dynamical terms. This will on the one hand forge a novel line of investigation in topological dynamics that intertwines finite approximation properties with asymptotic phenomena like mean dimension, and on the other establish broader connections between group actions and C*-algebras as part of the effort to understand general types of crossed products and their K-theoretic classifiability.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.