Enrico Berni: A Crash Course on Amenability

Amenability is a fundamental notion in group theory, consisting of the existence of a translation-invariant "average" functional on a group. Amenable groups were first introduced in the 1920s by John Von Neumann, while studying the so-called "Banach Tarski paradox", and nowadays are one of the most studied classes of groups in the literature, especially in the fields of geometric group theory, functional analysis and ergodic theory.

 In the course of the talk, we will give some characterizations of amenability, we will prove that abelian groups are amenable (a result due to Von Neumann, although the proof will be more modern), and we will see how the nonamenability of free groups, along with the the axiom of choice, is one of the main ingredients of the Banach-Tarski paradox.

 The talk will have minimal prerequisites, as every notion used will be properly introduced.

 For fans of: Geometry, Functional Analysis, Group Theory.