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I am interested to find out what are some real world applications that are "truly" modelled by Simplicial Complexes.

Note: There are many some real world applications like social networks / neural networks which are modelled by graphs/hypergraphs, but are made into simplicial complexes via processes like clique complex. For the purpose of this question, I would like to exclude the above types of applications, and focus on those which are already initially simplicial complexes, without going through the process of graph/hypergraph to simplicial complex.

It seems that by excluding the above applications, there are quite few "true" real world applications of simplicial complexes, in fact none that I know of. I would be grateful if someone could point out some.

Thanks a lot.

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    $\begingroup$ In 3D graphics, one of the main ways objects are modeled is by a map of a 2D simplicial complex to $\mathrm{R}^3$ (this is called a triangular mesh or sometimes a triangulation). $\endgroup$ Oct 4, 2017 at 19:05

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Not entirely sure if this satisfies you but there are ways of viewing neural networks as simplicial complexes that go beyond the traditional clique complexes. In a clique complex can fully connected subgraph, or clique, is considered a simplex in the simplicial complex, however if one imposes conditions on which cliques get chosen to be simplices and which get left as topological holes. Consider a neural network with various sized cliques, if all the neurons in a clique spike in-phase with each other that could be considered a simplex because we can think of that neuron group as acting as one unit. Contrarily, if there is a clique where the neurons are all spiking out of phase and not all that synchronized that can be left as a topological hole or n-dimensional cycle in the algebraic sense.

Giusti et al have a paper here, specifically figure 1 talks about the example I brought up. And also Reimann (not that one lol) et al have a paper discussing the different structural conditions to place on a neural network that dictate when certain cliques become simplices and some stay as holes.

Anyways, I think there a lot of real world examples that start as graph/hypergraphs but there exist a lot of ways to make different simplicial complexes out of the same data which I think can lead to different insights about hidden structure within these topological spaces.

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