Please note the unusual date and time for this talk!

The class of complete first-order theories where algebraic closure satisfies exchange (known as pregeometric theories) encompasses a wide range of theories, which extends the classes of strongly minimal and \(o\)-minimal theories, as well as many other well-known \(\textbf{NIP}\) examples such as \(p\)-adically closed fields. Outside of the \(\textbf{NIP}\) context, we also find many examples of pregeometric theories which are simple (such as pseudo-finite fields and supersimple theories of \(\textsf{SU}\)-rank 1), \(\textsf{NSOP}_4\) (such as Conant’s free amalgamation theories), and also neither of the two. However, there is a class of theories that has been intensely studied over the last decade for which no such examples are known: the class of \(\textsf{NSOP}_1\) theories.

In this talk, I will provide a partial result in this direction and show that, assuming the stable Kim-forking conjecture holds, there are no \(\textsf{NSOP}_1\) non-simple pregeometric theories with geometric elimination of imaginaries. The proof involves giving a partial answer to a question posed independently by Kim and d'Elbée in 2021 regarding the interaction between rosy and \(\textsf{NSOP}_1\) theories.