TITLE: Gromov's list of 7 constructions in differential topology
QUESTION [77 upvotes]: At the 2010 Clay Research Conference, 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.

REPLY [4 votes]: There is an interview with Gromov: http://www.ihes.fr/~gromov/PDF/rtx100300391p.pdf
Q: (...) an you describe your involvement and how your mathematical and geometric insights can be useful for problems in biology?
Gromov: I can explain how I got involved in that. Back in Russia, everybody was excited by ideas of René Thom on applying mathematics to biology. My later motivation started from a mathematical angle, from hyperbolic groups. I realized that hyperbolic Markov partitions were vaguely similar to what happens in the process of cell division. So I looked in the literature and spoke to people, and I learned that there were so-called Lindenmayer systems. (...)
And his paper on the subject: "Cell Division and Hyperbolic Geometry" http://www.ihes.fr/~gromov/PDF/16%5B71%5D.pdf
I was just reading: Visions in Mathematics: GAFA 2000 Special Volume, Part I.
Gromov's article in the collection, titled: "Spaces and Questions"
has a subsection: "Symbolization and Randomization" which you might find interesting, he discusses "random manifolds" at length and even touches on one of the questions in his talk: assembling combinatorial manifolds out of simplices (i.e. how many triangles).