turkmath.org

A subset of $\mathbb R^n$ is said to be semi-algebraic if it can be described by a finite system of polynomial equalities and inequalities. The collection of semi-algebraic sets enjoys stability by many natural operations (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 captures many of these features of semi-algebraic geometry. This axiomatic generalization 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 (though proving that these witnesses are indeed witnesses is a very difficult problem in general). I will give an overview of results concerning these two aspects of o-minimality (tame behaviour and natural examples) and if time permits try to explain how o-minimality gives perspectives on some aspects of the theory of ODE and how it also is used to prove some results in number theory.