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?

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    $\begingroup$ I would simply say that definable sets in o-minimal expansions of the real number field obey the desired properties outlined in the section on tame topology in the Esquisse, and hence could be regarded as the model-theoretic response to that section of the monograph. This is enunciated clearly in van den Dries's book Tame Topology and O-minimal Structures. $\endgroup$ – Todd Trimble Dec 9 '15 at 9:05
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    $\begingroup$ I was hoping for some more details, such as relation to semialgebraic and semianalytic sets as well as the status of current research in o-minimality with regard to the esquisse. $\endgroup$ – Mikhail Katz Dec 9 '15 at 9:13

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:

  • Every definable function $\mathbf R\to\mathbf R$ is piecewise monotone, piecewise $\mathbf C^k$ for every $k$. (No $\sin(1/x)$ curve, etc.)
  • The closure, the interior of a definable subset is definable (just by using the $\epsilon,\delta$ definition).
  • There is a cellular decomposition theorem. An open cell is a subset of $\mathbf R^n$ of the form, say, $$ \{(x,t)\in \mathbf R^{n-1}\times\mathbf R\, ;\, x\in A,  \phi_-(x) < t<\phi_+(x) \}$$ where $A$ is an open cell in $\mathbf R^{n-1}$ and $\phi_-,\phi_+$ are definable functions on $A$ such that $\phi_-(x)<\phi_+(x)$ for every $x\in A$. (Actually, there are other cells similarly defined by replacing strict inequalities by large inequalities, and taking $\phi_-=-\infty$ or $\phi_+=+\infty$.) Then for every finite family $\mathscr A$ of definable subsets of $\mathbf R^n$, there is a finite partition of $\mathbf R^n$ into cells such that each element of $\mathscr A$ is a union of some cells.
  • A closed definable subset of $\mathbf R^n$ is locally contractible (no hawaiian earring).
  • There is a nice theory of dimension (Brouwer invariance of domain is not an issue, for example).

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:

  • Semialgebraic geometry (where definable subsets are semialgebraic sets); by Tarki's theorem, every semialgebraic set can be defined without quantifiers, hence you see that semialgebraic subsets of the line are finite union of intervals.
  • Take for a basic family of functions the restrictions to $[-1,1]^n$ of functions defined by a power series of radius of convergence $>1$. This geometry is o-minimal (Denef, van den Dries). In particular, compact analytic subsets generates an o-minimal geometry which contains real subanalytic sets.
  • Take only one basic function, namely the exponential function (defined on the whole real line). This geometry is o-minimal (Wilkie).
  • Combine the two preceding geometries; you still get an o-minimal geometry (van den Dries, Macintyre, Marker) and this was a crucial tool in the recent solution of the André-Oort conjecture (Pila, Tsimerman, Klingler, Ullmo, Yafaev).
  • Speissegger shows you can add solutions of a definable Pfaff equation, etc.
  • $\begingroup$ The commands \\[ and \\] don't work at MO. $\endgroup$ – Mikhail Katz Apr 20 '16 at 14:46
  • $\begingroup$ The conclusion of Petezil-Starchenko is real-algebraic? $\endgroup$ – Mikhail Katz Apr 20 '16 at 14:48
  • $\begingroup$ non, complex algebraic! I'll edit! $\endgroup$ – ACL Apr 20 '16 at 14:57
  • $\begingroup$ I take it the collection of all definable functions between fixed source and target is also definable? This would answer Grothendieck's dream and should probably be mentioned in the answer. $\endgroup$ – Mikhail Katz Apr 20 '16 at 14:59
  • $\begingroup$ It is not clear to me that this requirement in Grothendieck's esquisse (“passing from $X$ to $\mathop{\rm Aut}(X)$ leaves the world of finite dimensional spaces”, he says) is satisfied by o-minimal geometry. At least not obviously. $\endgroup$ – ACL Apr 20 '16 at 15:01

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