At the [2010 Clay Research Conference][1], Gromov explained that we know of only 7 different methods for constructing smooth manifolds.  Working from memory, and hence not necessarily respecting the order he used:

 - Algebraic geometry (affine and projective varieties, ...)
 - Lie groups (homogeneous spaces, ...)
 - General position arguments (Morse theory, Pontryagin-Thom construction, ...)
 - Solutions to PDE (Moduli spaces in gauge theory, Floer theory, ...)
 - Surgery (Cut and paste techniques, ...)
 - Markov processes

I realise that I only gave 6 constructions; this was the number of separate items listed on his slides, and since he failed to discuss this part, I am left to guess that he either listed two different constructions on one line, which I interpreted to be variants of the same construction, or that failed to include one altogether.

**Question** How does one construct a smooth manifold from Markov processes?

I asked Gromov after the talk for explanation, but due to the rudimentary nature of my Gromovian, I was unable to understand the answer.  The only word I managed to parse is "hyperbolic," though I wouldn't put too much stock in that.

  [1]: http://www.ihp.jussieu.fr/Clay%2520Institut/Inscriptions/Clay%2520conference%25207-8-9%2520june%25202010.html%20%222010%20Clay%20Research%20Conference%22