The Banach-Tarski paradox shows that one can decompose a 3-dimensional ball into finitely many pieces and reassemble the pieces using rigid motions into two balls of the same size as the original, violating conservation of mass. It was proven using non-amenable groups: groups which admit paradoxical decompositions. The concepts of paradoxicality and amenability admit higher-order analogs called (non)-surjunctivity and soficity in which entropy plays the role of mass.
In these tutorials, I will give gentle introductions to amenable groups, classical entropy theory, sofic groups and the relatively new topic of sofic entropy theory.
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