Independence and Dichotomies In Dynamics and Operator Algebras
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The project will consist of a number of research and educational activities centered around the study of randomness, symmetry, paradoxicality, and dimensionality in dynamical systems. The goal is to understand the way that combinatorial, geometric, and topological phenomena interrelate within a newly broadened scope in dynamics that has been opened up by recent advances in entropy theory, the theory of infinite groups, and the classification program for nuclear C*-algebras. The investigations will be rooted in the combinatorial and probabilistic notions of independence and the various dichotomies to which they relate either as a source or as a counterpoint. Both the research and educational components of the program will bridge to several other branches of mathematics including geometric and combinatorial group theory, algebraic topology, convex geometry, and descriptive set theory. The mathematical theory of dynamics seeks to establish a rigorous framework for understanding the kinds of behavior that one might observe in a physical system as it evolves in time. In a chaotic system our ability to make predictions about the future state of the system is limited or null, and there are different ways in which this unpredictability can be made mathematically precise, for example via the notion of entropy. These concepts are all tied to the probabilistic notion of independence, which is exhibited in its most basic form by the repeated tossing of a coin. A general system will typically combine both deterministic and indeterministic aspects, often in an intricate way. This project aims to develop new tools for identifying the presence of independence in its various dynamical guises and its effect on associated mathematical structures such as operator algebras. On the practical side, the interest in such an endeavor lies in its significance for the study of problems in information theory involving code transmission and error correction.