Abstract：

The stable forking conjecture hypothesizes a sweeping connection between simplicity and stability, stating that every instance of dependence in a simple theory is exhibited by a stable formula. Even important special cases of this conjecture have long remained open: for example, it is open whether the independence relation is stable over any base, even in, say, a finite-rank supersimple theory. However, supersimple theories are sometimes tractable from the point of view of geometric stability theory, which has drawn deep connections between stability or simplicity and pregeometries: combinatorial geometries, such as the closure relation on an algebraically closed field. It has long been implicitly known that stable independence is determined by pregeometries in supersimple theories: for any n, there is a collection of pregeometries G_n such that, for any (finite-rank) supersimple theory T, the independence relation is stable over any base in T between rank-n types if and only if there is no minimal type in T whose pregeometry embeds one of the pregeometries in G_n. 

In this talk, we discuss our result that this relationship between the stable forking conjecture and pregeometries is not just superficial: this collection G_n, which determines stable dependence over a base in rank n, can be taken to be finite, so we have reduced stable dependence over a base in any finite rank to checking finitely many pregeometries. For n at most 3, this result follows more directly from work of Peretz on canonical bases for unstable forking sequences in rank 3, and Peretz hints in this work at generalizations to higher ranks. However, direct generalizations of Peretz's work break down in higher ranks, and our result appears to require a new twist on this work: showing, assuming this case of the stable forking conjecture fails, that some (rather than every) unstable forking sequence satisfies Peretz's conclusions.

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