What is a Manifold? - Mikhail Gromov
Why MathOverflow recommends it
MathOverflow users link it as a source for Gromov's views on constructing manifolds and for specific remarks they quote, such as triangulation counts and generic functions.
Recommended by 1 people across 1 different months, 2012 to 2016.
What it is
A 2010 Clay Research Conference lecture by Mikhail Gromov on what a manifold is and the many ways manifolds can be defined and constructed. It ranges over triangulations, generic objects and foundational remarks, and runs about 54 minutes.
What people said
Recommends David Roberts · 2016-03 · MathOverflow · on “What are the different ways of defining 3-manifolds?”
There was a good talk by Gromov called "What is a manifold" (see it here: [this video] ) where he discusses various ways we know how to get manifolds including some of the above. I think you have a few more that are specific to dimension 3, but his point is interesting in that there are a lot of 'generic' manifolds we really don't have good ways to construct and understand. Of course, with geometrisation, and virtual Haken etc conjectures settled, we have a much better grip for 3-manifolds than in general.
Cites Jonah Sinick · 2012-02 · MathOverflow · on “Asymptotics for the number of triangulations of a manifold M”
In Gromov's talk at the Clay Math Research [this video] from 23:23 to 25:55 Gromov says (slightly paraphrased) I want to emphasize a problem which comes from mathematical physics which is unsolved which is indicating that we don't understanding what triangulation is. And the problem is extremely simple. You take a manifold, and you just ask how many triangulations it has with a given number of simplices. So you have your manifold $X$ and you have the number of triangulations with k simplices $N_k(X)$ and you want to know what happens to it as $k$ goes to infinity, roughly. You take triangulations up to isomorphism. It's bounded below by $(1 + \epsilon)^k$ and it's bounded from above by $k^k$, rou
Cites Chill2Macht · 2016-08 · MathOverflow · on “Does anyone understand this comment about the continuum hypothesis?”
At 31:37 in his lecture titled What is a manifold? [this video] posted on Youtube, Mikhail Gromov states that if we do not allow generic functions to exist then the continuum hypothesis is "obviously" true, and that if we do allow generic functions to exist, then the continuum hypothesis is "obviously" false. What concept is he referring to? It sounds extremely important. Note: this is a follow-up to a question I posted on Math.SE: [LINK] Also I'm not sure how to tag this question, so please feel free to fix the tags as appropriate. EDIT: When Gromov mentions generic functions for the first time in the lecture, around 29:30 if I remember correctly, it seems like it is with regards to generic poin