Model theory and stability theory, with applications in differential algebra and algebraic geometry

Summary

This article is based around parts of the tutorial given by E. Bouscaren and A. Pillay at the training workshop at the Isaac Newton Institute, March 29 — April 8, 2005. The material is treated in an informal and free-ranging manner. We begin at an elementary level with an introduction to model theory for the non logician, but the level increases throughout, and towards the end of the article some familiarity with algebraic geometry is assumed. We will give some general references now rather than in the body of the article. For model theory, the beginnings of stability theory, and even material on differential fields, we recommend [5] and [8]. For more advanced stability theory, we recommend [6]. For the elements of algebraic geometry see [10], and for differential algebra see [2] and [9]. The material in section 5 is in the style of [7]. The volume [1] also has a self-contained exhaustive treatment of many of the topics discussed in the present article, such as stability, ω-stable groups, differential fields in all characteristics, algebraic geometry, and abelian varieties.

Model theory

From one point of view model theory operates at a somewhat naive level: that of point-sets, namely (definable) subsets X of a fixed universe M and its Cartesian powers M × … × M. But some subtlety is introduced by the fact that the universe M is “movable”, namely can be replaced by an elementary extension M′, so a definable set should be thought of more as a functor.