Alberto Miguel Gómez (Imperial College London) – Generalising Kim’s lemma to abstract independence relations

Recent developments in neostability theory have witnessed the proliferation of versions of Kim's lemma used to characterise ever-wider classes of unstable first-order theories, including NTP_2, NSOP_1, NSOP_4, NBTP, and beyond. These results highlight some limitations of Adler's framework for the study of abstract independence relations and suggest that new tools are at play underneath this diversity. In this talk, I will extend Adler's framework to study Kim's lemma and its consequences in their full generality. After briefly reviewing the role of independence relations in developing the theory of several neostability-theoretic properties, I will reinterpret Kim's lemma as a binary relation between independence relations under minimal assumptions. I will exemplify how this generalises several results from the literature, offer some new results that this framework proves, and mention previously known theorems which can be obtained in a semantic fashion using this technology.