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).



Why are discreteness and smoothness in physics inversed with respect to geometry?
The "manifold picture" can be applied to physics in the context of the [Brillouin zone](https://en.wikipedia.org/wiki/Brillouin_zone), see for example [On Brillouin Zones](https://www.math.stonybrook.edu/preprints/ims98-7.pdf). The reason that discreteness and smoothness appear inverted, is that the Brillouin zone describes [reciprocal space](https://en.wikipedia.org/wiki/Reciprocal_lattice). 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.


