The class of NSOP_1 theories properly contains the simple theories and is contained in the class of theories without the tree property of the first kind. We will describe a notion of independence called Kim-independence, which corresponds to non-forking independence ‘at a generic scale.’ In an NSOP_1 theory, Kim-independence is symmetric and satisfies a version of Kim’s lemma and the independence theorem. Moreover, these properties of Kim-independence individually characterize NSOP_1 theories. We will talk about what Kim-independence looks like in several concrete examples: parametrized equivalence relations, Frobenius fields, and vector spaces with a bilinear form. This is joint work with Itay Kaplan.
- Théorie des Modèles et Groupes