| Résume | A global invariant type p is called stable if there is no witness to the order property using realizations of p, and generically stable if there is no witness to the order property using a Morley sequence of realizations of p. Hrushovski and Pillay showed that in NIP theories, a type is generically stable if and only if it is definable and finitely satisfiable in some small model. These latter properties generalize readily to Keisler measures, and so this result laid the foundation for subsequent work of Hrushovski, Pillay, and Simon on generically stable measures in NIP theories. In particular, they showed that a generically stable measure in an NIP theory is uniformly and almost surely interpreted by frequency measures, i.e., a "frequency interpretation measure". Outside of the NIP setting, this characterization no longer holds, which leads to competing options for the "right" definition of generic stability for Keisler measures in general. The focus of this talk will be on comparing and contrasting these various notions, starting with an earlier result with Gannon on "frequency interpretation types". I will then present several recent positive and negative results on the behavior of Keisler measures in arbitrary theories, as well as some examples and counterexamples. This is joint work with Gannon and Hanson. |
