The notion of o-minimality comes from model theory, was introduced by Anand Pillay and Charles Steinhorn in 1986. O-minimal structures are structures (of a certain language) endowed with a total (dense) ordering, and having the following property: every definable subset of the line is a finite union of intervals and points. Starting with this very simple property, one shows results of uniformity, cell decompositions, etc.
O-minimal structures have shown their importance in many other domains of mathematics: in real and analytic geometry, in number theory (Pila-Wilkie and its applications), and more recently in combinatorics.
Important examples are the ordered field of real numbers (R,+,×,0,1,<), possibly enriched with the exponential and analytic functions restricted to compact sets.
The course will start with the basic properties of the field of real numbers: Theorem of Tarski-Seidenberg (the projection of a set defined by polynomial equations and inequalities is of the same form). Ordered fields, existence of real closures, dimension, etc.
We will then study o-minimal structures and their properties. (Our study will concentrate on fields, in particular on the field of real numbers with additional structure).
Definition of o-minimality
Basic properties
Cellular decomposition
Vapnik-Chervonenkis property and associated invariants
We will then pass to applications:
The field of real numbers with restricted analytic functions and the full exponential is o-minimal.
Reparameterization and the theorem of Pila- Wilkie
Definability of the exponential function
Other applications
Prerequisites: Some knowledge of basic model theory will be helpful (but not absolutely necessary; see my course notes on my web page: chapters 2 and 3). Algebra, and in particular field theory.