turkmath.org

A subset of $R^n$ is said to be semi-algebraic if it can be described by a finite system of polynomial inequalities. The collection of semi-algebraic sets enjoys stability by many natural operation (taking the topological closure, projecting,...) making it a rich tool to use, and at the same time its objects satisfy geometric theorems (stratification in smooth manifolds,...) that allow to keep control on their behaviour. O-minimality is an axiomatic approach which capture many of these features of semi -algebraic geometry. This axiomatic generalisation is successful in at least two sense: it preserves most of the nice properties of semi-algebraic geometry, and at the same time it has many natural witnesses that allow to "export" these properties outside of the algebraic world. I will give an overview of results concerning these two aspect of o-minimality.