Lecture Notes -stability Theory (math 414) Spring 2003

The aim of this course and these notes is to present an exposition of the basics of stability theory, stable group theory, and geometric stability theory. I will assume knowledge of my Autumn 2002 model theory lecture notes [1]. In fact the current notes are a natural continuation of the earlier ones. The model theory lecture notes ended with a proof of Morley’s Theorem. Stability theory developed historically (in the work of Shelah) as a chunk of machinery intended to help generalize Morley’s theorem to a computation of the possible “spectra” of complete first order theories. Here the spectrum of T is the function I(T,−), where I(T, κ) is the number of models of T of cardinality κ. This project (at least for countable theories) was essentially completed by Shelah around 1980. In the meantime other perspectives developed, in which stability theory is seen rather as a way of classifying definable sets in a structure and describing the interaction between definable sets. Eventually this theory was seen as having a “geometric meaning”. This means, on the one hand, that various structural results have a geometric flavour. On the other hand, it refers to the empirically discovered fact, that in numerous examples, the model-theoretic notions have “actually existing” mathematical meaning. This will be the perspective of these notes. We have a choice of the level of abstraction at which to operate. The theory of forking can be developed smoothly for the class of simple theories. There have been recent developments, such as rosy theories which subsume stable theories, most simple theories, and 0-minimal theories. ω-stable (or