Wolfram Cloud Resource Object

Paclet Resource

WolframInstitute/DiscreteGeometry

Discrete geometry of combinatorial objects

Contributed By: Nikolay Murzin, Pavel Hajek

DiscreteGeometry is geometry and topology through complexes built from a graph. The objects are clique complexes, Delta-complexes and simplicial sets; the clique complex of a graph is its Vietoris-Rips complex at radius 1. On a complex the paclet computes the chain complex and its homology, the Euler characteristic, curvature, and the Hodge and Dirac operators.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall[ResourceObject["https://wolfr.am/1HTxRtd9x"]]


To load the code after installation, evaluate this code:
Needs["WolframInstitute`DiscreteGeometry`"]

Details

A complex is a list of simplices, each a list of vertices. A complex built from a graph is closed under taking faces; a Delta-complex or a simplicial set also records how its faces are glued.
The chain complex of a complex is its list of boundary matrices. Homology, Betti numbers, the Euler characteristic and the norms of homology classes are read off it.
The Hodge Laplacian and the Dirac operator act on the chains of a complex. The connection and Green function matrices, and the Forman-Ricci and Lefschetz curvatures, live on the same complex.
Further subjects: hypergraphs and their complexes, meshes and embeddings of complexes, quantum calculus on a complex (waves, Dirac walks, zeta functions, analytic torsion), and topological data analysis (Vietoris-Rips filtrations and persistent homology).
Metric constructions, such as balls, volume growth and Ollivier curvature, are in the sibling paclet InfraGeometry.
Install with PacletInstall[ResourceObject["https://www.wolframcloud.com/obj/hajek_pavel/DeployedResources/Paclet/WolframInstitute/DiscreteGeometry"], ForceVersionInstall → True], then load with Needs["WolframInstitute"].

Examples

Paclet Source