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 !

  • $\begingroup$ 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

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.