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I wonder if mathematicians have developed a discrete version of non-commutative geometry, a bit like graphs, simplicial complexes etc may be seen as a discrete version of (Riemannian) geometry (of course with its own richness and non reductibility to smooth geometry).

If someone could give me basic references on the topic, I would be glad to dive into them (and come back with questions, for sure !).

Thanks beforehand !

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  • $\begingroup$ you might be interested in a paper from Majid arxiv.org/abs/1011.5898 Noncommutative riemannian geometry on graphs. This is not NCG in the manner of connes, though. $\endgroup$
    – mebassett
    Dec 31, 2013 at 7:44
  • $\begingroup$ Maybe check out Marcolli, particularly the book Arithmetic Noncommutative Geometry, or her paper Spin Foams and Noncommutative geometry its.caltech.edu/~matilde/SpinFoamCover.pdf $\endgroup$
    – user3146
    Sep 4, 2019 at 7:14

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There's nothing really systematic, but there is a fair bit of literature already on what you might call discrete noncommutative Riemannian geometry à la Connes, dealing with noncommutative spectral triples that are discrete in some sense.

On the one hand, there's been a great deal of work on so called "finite" or "discrete" spectral triples, spectral triples corresponding to noncommutative spaces with metric dimension $0$ but possibly non-zero $K$-theoretic dimension ("$KO$-dimension"); these have been used as $0$-dimensional "internal spaces" in noncommutative-geometric particle physics models. The seminal papers on finite spectral triples and their classification are by T. Krajewski and by M. Paschke and A. Sitarz, and the most up-to-date survey on their standard applications within noncommutative-geometric mathematical physics, including a great deal of introductory material, is by K. van den Dungen and W. van Suijlekom. For a particularly interesting recent development, take a look at M. Marcolli and W. van Suijlekom on connections with spin networks and lattice gauge fields.

On the other hand, there's been a great deal of work on constructing interesting spectral triples for graphs and fractals, though I must confess to knowing little about this line of work. A literature trawl brings up potentially interesting papers by M. Requardt, E. Christensen, C. Ivan, and M. Lapidus, and J. W. de Jong, for instance. There's also a fair bit of work on semi-finite spectral triples for graph $C^\ast$-algebras, by A. Rennie and collaborators, but this I know even less about.

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