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In a closed (say differentiable) Riemannian manifold you see only continuous features when looking at small neighbourhoods of points. From afar, discrete features appear ((co)homology, closed geodesics, eigenvalues of the laplacian and so on).

In physics, I have the impression that more or less the opposite is going on: seen from afar you get (general) relativity, differential equations etc. Zooming in you have particles, quantum mechanics, discrete eigenvalues, ....

An obvious explanation might be that the 'manifold-picture' is not a good model for our universe.

Is there an obvious mathematical object which behaves like the universe: local properties are mainly discrete and global properties tend to be continuous? (Coming up with the adele-stuff is kind of cheating, I think, since 'locality' in this context is mainly a convention. The other obvious 'explanation', string-theory, seems also to be somewhat controversial.)

Which mathematical objects behave like the universe with respect to smoothness/discreteness?

Motivation: I have no real motivation for this question other than curiosity: It is quite striking that one of the most studied objects of mathematics, manifolds, has smoothness/discreteness inversed with respect to physics (in my limited understanding of both areas).

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    $\begingroup$ Sure: Consider $M$ equal to all integer translates of a Cantor subset of $[0,1]$ and equip $M$ with the distance $d(x,y)=|x-y|$. As you zoom in, everything is totally disconnected ("discrete" in your terminology, I assume), when, as you zoom out, in the (Gromov-Hausdorff) limit you recover geometry of the straight line. $\endgroup$
    – Moishe Kohan
    Commented Jun 14, 2022 at 12:41
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    $\begingroup$ @MoisheKohan Not a bad example (by the way, many similar examples arise from geometric group theory, e.g. fundamental groups of hyperbolic manifolds), but I feel somehow cheated: Things are not as 'natural' as in the manifold-case and our Universe seems quite natural to me. $\endgroup$
    – Roland Bacher
    Commented Jun 14, 2022 at 12:57
  • $\begingroup$ Ok, then tell me what do you mean by "natural": I find my example natural. $\endgroup$
    – Moishe Kohan
    Commented Jun 14, 2022 at 13:19
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    $\begingroup$ Very interesting question! I know next to nothing about it, but in physics there is a host of different kinds of dualities, including ones interchanging small and large scales. On the other hand there are some dualities in mathematics that might be similar in spirit to that. Back in 2017 I asked a question about possible relevance of the Spanier-Whitehead duality for physics but it did not produce much feedback, except for Aaron Bergman mentioning that possibly Alexander duality is more or less physicist's duality between charges and fields... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jun 14, 2022 at 13:24
  • $\begingroup$ @MoisheKohan Kantor sets are obtained by removing pieces recursively and this has the same feeling as the 'It's turtles all-the-way-down'-thing. But perhaps I am wrong and the Universe is indeed 'turtling down'. $\endgroup$
    – Roland Bacher
    Commented Jun 14, 2022 at 13:34

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The "manifold picture" can be applied to physics in the context of the Brillouin zone, see for example On Brillouin Zones. The reason that discreteness and smoothness appear inverted, is that the Brillouin zone describes reciprocal space. The distinction is not fundamental, one can equivalently describe a crystal in real space, where discrete features appear on short distance, or in reciprocal space, where discrete features appear at large distance.

So to answer the specific question in the OP: I don't think there is a need to abandon the manifold picture to describe physical matter, you just want to apply it to reciprocal space rather than to real space.

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  • $\begingroup$ We could brush up on this and try to figure out in which way physics and math are "dual to each other". One keyword that comes to my mind is Wick rotation (hence an involution, so a possible good candidate for the wished kind of duality), establishing a parallel between "physics in euclidean metric signature" and math. My idea is that thermodynamical systems at equilibrium are very similar to purely mathematical ones, where things don't change with time. $\endgroup$
    – Sylvain JULIEN
    Commented Jun 14, 2022 at 14:49
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    $\begingroup$ Wick rotation? I would think Fourier transformation; manifolds appear upon Fourier transformation from real space to reciprocal (or momentum) space; this is of course also an involution, but Wick rotation is from real time to imaginary time, which I don't think helps here. $\endgroup$
    – Carlo Beenakker
    Commented Jun 14, 2022 at 16:31
  • $\begingroup$ Imaginary time is equivalent to space, hence induces a loss of temporality. $\endgroup$
    – Sylvain JULIEN
    Commented Jun 14, 2022 at 20:55
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    $\begingroup$ I think that's a very good answer. $\endgroup$
    – Konrad Waldorf
    Commented Jun 17, 2022 at 7:24

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