Michael Björklund: Fieldwork: Adventures in Ergodic Multiplication

Abstract:

Given a countably infinite field, one can naturally associate to it a dynamical system via the action of its multiplicative group on the Pontryagin dual of its additive group, equipped with the Haar probability measure. This construction yields a probability measure-preserving system that is 2-mixing but fails to be 3-mixing. In this talk, we explore the extent to which this dynamical system retains information about the underlying field. This question leads us to consider Kloosterman-type exponential sums over general (not necessarily finite) fields—an area that appears to be largely unexplored. Particular difficulties arise when the field is not a union of finite subfields, in contrast to the classical setting. I will discuss these challenges and present recent joint work with A. Fish (University of Sydney).