This is on joint work with John Baldwin and James Freitag. Shelah
showed that stable theories, a class of first-order theories defined
entirely in terms of the combinatorics of individual formulas, can
nonetheless be understood structurally, by generalizing independence
phenomena from all throughout mathematics. Since Shelah’s work,
there’s even been success in using the independence relation from
stability theory to describe unstable theories, including simple
theories and theories without the first strict order property (NSOP_1
theories). The stable forking conjecture, one of a small handful of
the most important problems in model theory, asks whether this
connection to stability theory is more than just a structural one.
Specifically, it asks whether the independence relation in simple
theories, beyond just sharing many properties with the independence
relation in stable theories, is determined by actual stable formulas.
We propose a version for NSOP_1 theories, the simple Kim-forking
conjecture, and discuss how it is amenable to some new approaches; for
example, as a consequence of work of Kaplan and Ramsey (2021),
a global variant of the simple Kim-forking conjecture is true in
general.