Amenability in action

It is said that groups, just like people, will be best known by their actions.

This is the guiding principle of this proposal, which focuses on the study of one of the most ubiquitous notions in mathematics: amenability.

Since its introduction by von Neumann in the context of the Banach-Tarski paradox, amenability has seen unexpected connections to numerous areas of mathematics.

This research proposal, which is roughly divided into four projects, takes another step towards understanding group amenability, by studying it through the lenses of group actions on C*-algebras.The first two projects explore the tension between amenability and its counterpart paradoxicality in the setting of C*-dynamical systems.

We regard amenability as a malleability property, which is known to lead to several uniqueness-type results.

On the other hand, paradoxicality is expected to produce wild behaviors from the point of view of classifiability and Borel complexity.

The last two projects deal exclusively with actions of amenable groups, and aim at establishing a pair of recent conjectures: one in topological dynamics, due to Kerr, and another one in C*-dynamics, due to the PI and Lupini. The new results on Folner tilings, combined with novel techniques, are expected to be of great aid in this project.