This is the title of a seven-parts talk given at the CMU Model Theory Seminar. The talks started on Oct. 13, 2014 and ended on Dec. 1, 2014.
In 1990, Makkai and Shelah studied the class of models of an Lκ,ω sentence, where κ is strongly compact. Among many other results, they showed that Galois types (a purely semantic notion of types) and syntactic types conveyed the same information. In particular, Galois types are determined by their restrictions to sets of size less than κ. This last property was later isolated by Grossberg and VanDieren and called tameness. In this talk, I will show that tameness already implies that Galois types are (in some sense) syntactic, thus generalizing Makkai and Shelah's result. I will give several applications to the stability theory of tame abstract elementary classes.
Michael Makkai, Saharon Shelah. Categoricity of theories in Lκ,ω, with κ a compact cardinal, Annals of Pure and Applied Logic 47 (1990), 41-97.
Sebastien Vasey, Forking and superstability in tame AECs, Submitted. Preprint: pdf arXiv.
Anand Pilay. An introduction to stability theory, Dover, 1983.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 18 and 20, College Publications, 2009.