Possibilities for a theory of independence beyond NSOP_1 and NTP_2
Non-forking independence was introduced by Shelah in Classification Theory and it later served as the foundation for simplicity theory, developed by Hrushovski, Kim, Pillay, and many others.  One of the key features of non-forking independence in simple theories is Kim's lemma:  in a simple theory, a formula divides over a set if and only if it divides with respect to some or equivalently all Morley sequences in the parameters.  This property was subsequently generalized in two very different directions.  The first, in the context of NTP_2 theories, was established by Chernikov and Kaplan, who showed that the same conclusion holds in NTP_2 theories, provided Morley sequences are replaced by strict invariant Morley sequences.  Secondly, in the context of NSOP_1 theories, we show that the same conclusion holds in NSOP_1 theories, with "dividing" replaced by Kim-dividing.  We will describe some work in progress with Alex Kruckman that describes a unifying context for both of these classes of theories, tied together by a variant of Kim's lemma and incorporating many new examples.