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.