Independence in tame abstract elementary classes

This is the title of a talk given at the model theory seminar at the City University of New York on Oct. 31, 2014. The talk presents the corresponding paper. Here are slides from a similar talk in South Korea.


Good frames are one of the main notions in Shelah's classification theory for abstract elementary classes. Roughly speaking, a good frame describes a local forking-like notion for the class. In Shelah's book, the theory of good frames is developped over hundreds of pages, and many results rely on GCH-like hypotheses and sophisticated combinatorial set theory.

In this talk, I will argue that dealing with good frames is much easier if one makes the global assumption of tameness (a locality condition introduced by Grossberg and VanDieren). I will outline a proof of the following result: Assume K is a tame abstract elementary class which has amalgamation, no maximal models, and is categorical in a cardinal of cofinality greater than the tameness cardinal. Then K is stable everywhere and has a good frame.