Idempotent Keisler measures – Artem Chernikov


In model theory, a type is an ultrafilter on the Boolean algebra of definable sets, and is the same thing as a finitely additive {0,1}-valued measure. This is a special kind of a Keisler measure, which is just a finitely additive real-valued probability measure on the Boolean algebra of definable sets. If the structure we are considering expands a group (i.e. the group operations are definable), it often lifts to a natural semigroup operation on the space of its types/measures, and it makes sense to talk about the idempotent ones among them. For instance, idempotent ultrafilters on the integers provide an elegant proof of Hindman’s theorem, and fit into this setting taking the structure to be (Z,+) with all of its subsets named by predicates. On the other hand, in the context of locally compact abelian groups, classical work by Wendel, Rudin, Cohen (before inventing forcing) and others classifies idempotent Borel measures, showing that they are precisely the Haar measures of compact subgroups. I will discuss recent joint work with Kyle Gannon aiming to unify these two settings, leading in particular to a classification of idempotent Keisler measures in stable groups and further results on NIP.