Forking and thorn-forking on generically stable types

Darío García, University of Leeds. Part of the logic seminar series.

Abstract notions of "smallness" and independence relations related to them are among the most important tools that model theory offers for the analysis of arbitrary structures.
Two of the most important notions of smallness in model theory are forking (which captures "algebraic" independence in stable and simple theories, and has been understood to correspond to certain measure 0 ideals on NIP structures), and thorn-forking, which in a sense captures the notion of "topological" or "analytic" dimension in many important cases. Under certain mild assumptions, forking is the finest notion of smallness, whereas thorn-forking is the coarsest.
In stable theories these notions coincide. This is a nontrivial fact, which may have interesting consequences. For example, in the theory of algebraically closed fields, this corresponds in a sense to the equality of algebraic and analytic dimensions in complex algebraic geometry. It is therefore a natural and interesting question to ask under which conditions forking and thorn-forking agree, and the consequences that can be concluded when the two notions coincide.
In this talk I will give an overview of thorn-forking and generically stable types, and show that for generically stable types the notions of forking and thorn-forking agree. If time permits, we will explore some applications of this equivalence.