I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So in broad terms, what is the precise connection between o-minimal models and Grothendieck's programme?
Relationship Between O-Minimal Theory and Grothendieck’s Program
ag.algebraic-geometrylo.logicmodel-theoryo-minimal
Best Answer
To define an o-minimal geometry, one gives oneself a family of functions from $\mathbf R^n$ to $\mathbf R^m$ ($m,n$ not specified), and one considers all definable subsets of $\mathbf R^n$, ie, those which can be defined by a mathematical expression using these basic functions, addition, multiplication, constant functions, the ordering symbol, logical connectives (AND, OR, NOT) and quantifiers (FORALL, EXISTS). A function is called definable if its graph is definable.
The geometry is said to be o-minimal if the only definable subsets of $\mathbf R$ (the real line) are finite unions of intervals.
These geometries are tame in the sense of Grothendieck's Esquisse, but what is remarkable is that the tameness axiom is on subsets of the line, not of higher dimensional $\mathbf R^n$. Here are a few tame properties of o-minimal geometries:
As indicated by Todd Trimble, the book of van den Dries, Tame Topology and O-minimal Structures explains this in quite a detail. Also, a remarkable theorem of Peterzil and Starchenko states that a complex analytic subspace of $\mathbf C^n$ which is definable (when you identify $\mathbf C^n$ with $\mathbf R^{2n}$) is automatically complex algebraic.
It is also a fundamental fact that there are many interesting examples of o-minimal geometries. Let's me list some of them: