Spec Z analogue of Thurston program? It's been known for a while that primes in number fields can be thought of, from an algebraic point of view, to be similar to knots in 3-manifolds. A good reference (thanks to this question) would be an article by Morishita, 0904.3399. 
There are therefore many good analogues of operations, such as covers, or objects, like zeta-functions, that are defined purely algebraically. For example, a linking number of two knots has an easy algebraic definition as the image of one knot in the homology of the complement to the other which is analogous to residue symbol in number theory.
However, the operations of taking connected sum and cutting/gluing along a subsurface don't appear immediately to have an analogue in number fields. If you know how to make sense of "gluing" two schemes $\operatorname{Spec} \mathcal{O}_K$ and $\operatorname{Spec} \mathcal{O}_L$ along the "common element $x \in K, L$, by all means, please tell us!
Either way, here's my question:

What could be an analogue of the Thurston geometrization program for number fields?

(may be this analogue will not be using gluing-like operations after all?)
 A: I don't think there is any reason to think that one exists, especially because the analogy is not very tight. For example, if $X_K = \operatorname{Spec}(\mathcal{O}_K)$, there exist closed hyperbolic 3-manifolds $M$ such that the abelianization of the fundamental group is infinite. (In fact, one conjectures that all hyperbolic $M$ virtually (= after passing to a finite cover) have this property.) On the other hand, the abelianization of $\pi_1(X_K)$ is always finite, by class field theory. As has been remarked elsewhere, there are several non-trivial $K$ such that $pi_1(X_K)$ is trivial.
A: McMullen's third lecture at the 2000 Washington DC AMS Colloquium is exactly addressing this question.  See his slides at
http://www.math.harvard.edu/~ctm/expositions/home/text/talks/ams/dc00/html/index.html
A: I was surprised that there exist even arithmetic analogies to solitons (more) and Laumon's arithmetic version of an idea of Witten made a new proof of Weil II possible. What else may have arithmetic versions? Ricci flow?
On the Ricci flow-renormalization issue in the comments below, Urs Schreibers answer, an other expert yesterday:
"The renormalization that is involved is not the same as in QFT, except for the fact that it can also be thought as realizing, in that geometric context, a subtraction of divergences
that has the effect of keeping the flow solutions from blowing up. Whether there is in that context any role for algebraic structures of renormalization, such as Hopf algebras accounting for nested divergences, is a good question."
