Daniel Max Hoffmann: Convolution of Keisler measures and subgroups of Aut(M)

Abstract

In the talk, I plan to explain the main goals and results from my project with Kyle Gannon and Krzysztof Krupiński. We introduce a new product of types and Keisler measures, which is closely related to the action of \(\operatorname{Aut}(M)\) on the space of types. In fact, our idea for the product comes from an isomorphism theorem which we obtained at the beginning of the project. In the theorem, we shift the structure of an Ellis semigroup to some space of Keisler measures (i.e. regular Borel probability measures on a space of types). After that, we generalize the definition of the product and call it *-product. The semigroup of types equipped with *-product is interesting on its own and encodes important properties of the theory, but in my talk I will focus more on the main conjecture from our project, which - roughly speaking - states that there is a bijective correspondence between idempotent Keisler measures (idempotent with respect to the *-product) and closed subgroups of \(\operatorname{Aut}(M)\). We already know that this conjecture holds in all stable theories, in NIP theories with an additional KP-invariance assumption on the measures, and for types in all rosy theories.