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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.

S. Carnahan
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Mohammed Abouzaid
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    Stupid naive question: Where does covering spaces, open book decompositions, triangulations, etc. fit in? If it's Morse theory, then isn't surgery Morse theory as well? – Daniel Moskovich Jun 21 '10 at 13:46
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    I personally would put "covering spaces, open book decompositions, triangulations" into "cut and paste" and therefore "surgery". – Deane Yang Jun 21 '10 at 14:43
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    Hi Mohammed, perhaps the Dunfield-Thurston random 3-manifolds are examples? http://arxiv.org/abs/math/0502567 Take an $N$-step random walk on a Cayley graph of the mapping class group of a hyperbolic surface, so as to produce a random mapping class; glue two handlebodies to get a random 3-manifold with a Heegaard splitting. This is surgery, but the randomness highlights certain features (random 3-manifold fundamental groups have many finite-index subgroups compared to groups with random balanced presentations). – Tim Perutz Jun 21 '10 at 15:40
  • I would add bundles as another way to construct manifolds. Looking at your list, I don't don't think this fits into any category (of course, I wasn't at the talk). I wouldn't be surprised if Gromov mentioned bundles in his talk, since he has put them to good use. – Ian Agol Jun 26 '10 at 17:59
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    According to my notes, the 7 constructions Gromov listed are: triangulations & surgery; Lie groups and locally homogeneous spaces; algebraic equations; genericity & transversality; partitions and Markov spaces; solutions of elliptic variational problems; and moduli spaces. – Maxime Bourrigan Sep 23 '10 at 11:29
  • The website of the IHP has been completely re-designed, the end-result being that I was unable to find Gromov's slides anywhere (the best I could do is the abstract on the Clay site: http://www.claymath.org/researchconference/2010/gromov-abstract.txt ). If someone can track down a link and add it, I'm sure it would be useful. – Thierry Zell Sep 23 '10 at 12:21
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    +1 for `the rudimentary nature of my Gromovian'. – HJRW Sep 23 '10 at 14:39
  • The video of his talk is now available from the CMI website (updated link again): http://www.claymath.org/library/video-catalogue – j.c. Apr 18 '18 at 18:01
  • Gromov seems to have defined a notion of "Markov compactum" -- a kind of inverse limit of compact simplicial complexes -- and predicted that the boundary of a hyperbolic group $\Gamma$ is always a "Markov compactum". This was worked out by Dominika Pawlik here: https://arxiv.org/abs/1503.04577 . I don't know why Markov compacta are so named, and I suspect that they are manifolds if and only if they are spheres, which at least conjecturally implies that $\Gamma$ is commensurable to the fundamental group of a hyperbolic manifold.... – HJRW Apr 19 '18 at 10:00
  • ... So Gromov's Markov compacta are interesting spaces, but not really interesting manifolds. – HJRW Apr 19 '18 at 10:01

3 Answers3

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Unfortunately I missed the talk, but on the other hand Gromov have just produced a new paper called

Manifolds : Where do we come from ? What are we ? Where are we going ?

It can be found on his web page. From the title I guess there could be some intersection with the talk. In particular in section 11 called Crystals, Liposomes and Drosophila Gromov is speaking about "Markov quotients". This sounds like a way to produce "spaces" (generalisation of manifolds, I guess).

http://www.ihes.fr/~gromov/PDF/manifolds-Poincare.pdf

Dmitri Panov
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I suspect (but am far from certain) that Gromov may be referring to the correspondence between symbolic and hyperbolic dynamics.

The idea is basically that the 0-1 matrix corresponding to the sparsity pattern of a stochastic matrix encodes a subshift of finite type or topological Markov chain. Usually, however, one goes from the hyperbolic dynamics to the Markov description via a Markov partition or section.

I am not aware of a way to go in the other direction in general, although placing certain conditions on the Markov process would facilitate the construction of a Markov partition (which can then be made as small as one likes), for which covering sets would constitute an atlas.

Update: So I did a little digging and came across a paper by Coornaert and Papadopoulos called "Symbolic coding for the geodesic flow associated to a word hyperbolic group" (Manuscripta Math. 109, 465–492 (2002), DOI 10.1007/s00229-002-0321-9, PDF available here). In it the authors discuss an idea of Gromov whereby a to each "word hyperbolic group" $\Gamma$ a space with a flow defined up to orbit equivalence is given: this flow is called the geodesic flow associated to $\Gamma$. I quote:

In the case where $\Gamma$ is the fundamental group of a compact Riemannian manifold $M$ of negative curvature, then $\Gamma$ is word hyperbolic and [the geodesic flow associated to $\Gamma$] is, up to orbit equivalence, the geodesic flow on the tangent bundle of $M$.

Nowhere, however, is it indicated that the space so constructed is generically a manifold. Still, this construction is quite closely associated with the ideas mentioned earlier, as the introduction to this paper points out.

Steve Huntsman
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  • http://mathoverflow.net/questions/8916/a-regularity-property-of-transition-matrices-for-the-cat-map – Steve Huntsman Jun 21 '10 at 14:05
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    Any chance you could elaborate on your answer and explicitly say where a manifold appears? – Deane Yang Jun 21 '10 at 14:45
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    Although the geodesic flow of a hyperbolic group is mentioned frequently in Gromov’s seminal article on hyperbolic groups, nobody has managed to reproduce what he had in mind. The theorems he predicted were proved by other means. Eventually, Igor Mineyev produced a construction of the geodesic flow for an arbitrary hyperbolic group. It’s a very wild object; I don’t think it ever gives a manifold unless you start with one. – HJRW Apr 18 '18 at 19:36
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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).

devnull
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