The class of NSOP1 theories, originally introduced by Džamonja and Shelah in 2004, has been studied very intensively in the last few years since the striking discovery of an independence relation called Kim-independence by Ramsey and Kaplan (based on earlier ideas of Kim and a work of Chernikov and Ramsey), which generalises forking independence in simple theories, and retains all its nice properties except base monotonicity in the class of NSOP1 theories (over models).

I will start the talk with an overview of the notion of Kim-independence, and then I will present my joint work with Mark Kamsma on generalising Kim-independence to positive logic. In particular, I will discuss examples of positive NSOP1 theories falling into our framework such as the positive theory of existentially closed exponential fields (studied by Kirby and Haykazyan), and the hyperimaginary extensions of NSOP1 theories.