Notes on o-Minimality and Variations

Summary

The article surveys some topics related to o-minimality, and is based on three lectures. The emphasis is on o-minimality as an analogue of strong minimality, rather than as a setting for the model theory of expansions of the reals. Section 2 gives some basics (the Monotonicity and Cell Decomposition Theorems) together with a discussion of dimension. Section 3 concerns the Peterzil-Starchenko Trichotomy Theorem (an o-minimal analogue of Zil'ber Trichotomy). There follows some material on definable groups, with powerful applications of the Trichotomy Theorem in work by Peterzil, Pillay and Starchenko. The final section introduces weak o-minimality, P-minimality, and C-minimality. These are analogues of o-minimality intended as settings for certain henselian valued fields with extra structure.

1. Introduction

This paper is a survey of selected topics in and around o-minimality. The emphasis is on analogies with stability theory, and there is little here on analytic expansions of the reals. On the other hand, there is quite a lot on dimension in o-minimal and related structures. I have concentrated on algebraic examples. Section 2 is introductory, and covers definable functions, cell decomposition, dimension for definable sets, prime models, and definable types. Section 3 is on groups definable in o-minimal theories: there I describe a trichotomy theorem due to Peterzil and Starchenko, and consequences, due to them and Pillay, for definable groups. In Section 4, I leave o-minimality and turn to other settings (weak o-minimality, C-minimality, P-minimality) which are superficially similar, and survey some of the main examples and results.