An introduction to $C$-minimal structures and their cell decomposition theorem

Abstract

Developments in valuation theory, especially the study of algebraically closed valued fields, have used the model theory of $C$-minimal structures in different places, e.g., the work of Hrushovski-Kazdhan in [5] and Haskell-Hrushovski-Macpherson in [3]. We intend with this text both to promulgate a basic comprehension of $C$-minimality for mathematicians interested in valuation theory (equipped with a basic knowledge of model theory), and to provide a slightly different presentation of the cell decomposition theorem proved by Haskell and Macpherson in [6].