Entropy beyond actions of amenable groups

Entropy beyond actions of amenable groups

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Dylan Airey, Princeton University
Fine Hall 110

In 1959 Kolmogorov and Sinai introduced entropy for measure preserving transformations of a probability space, generalizing the notion from statistical mechanics and information theory. Their definition can be naturally extended to measure preserving actions of amenable groups, groups which one can average over in a suitable sense. On non-amenable groups, e.g. the free group with at least two generators, this definition breaks down. In fact an example of Ornstein and Weiss from 1987 cast doubt on whether a useful notion of entropy could be defined beyond amenable groups. However in 2010 Lewis Bowen developed an entropy theory for actions of a class of groups called sofic groups, which properly include amenable groups. In this talk I will first discuss the classical theory of entropy and its  various interpretations, then compare it to sofic entropy. Time permitting I'll mention some open problems in the area and connections to other areas like operator algebras. Focus will be put on specific examples and developing intuition. No knowledge beyond basic analysis will be assumed.