Hodge.jl

One of the most powerful tools available to study the algebraic topology of manifolds is the Hodge decomposition. Recently, a discrete analogous of it has been successfully applied for better understanding how one may transform a pairwise ranking that is cyclically inconsistent into some kind of global ordering.

The Julia package Hodge.jl is a library implementing the necessary tools of computational algebraic topology to easily utilize the Hodge decomposition. The available structures are representations for simplicial complexes and the algebra of cochains (discrete analogues of differential forms) as well as methods for calculating Betti numbers and the Hodge decomposition.

This package exports two main types, SimplicialComplex and Cochain, together with methods to work with their topological and algebraic properties.

The topological operations on this package are all done via the discrete Laplacian operator. This includes the method betti, which calculates the Betti numbers of a simplicial complex, and the method hodge which calculates the discrete Hodge decomposition of a cochain.

Installation

You can install this package via the Julia Package Manager. Simply open the REPL, enter ] and run

pkg> add Hodge

Bibliography

Hodge.jl is based on a scientific initiation that I did with Prof. João Paixão while an undergraduate at UFRJ.

The simplicial complex type is built upon the Simplex Tree data structure, described in the paper:

• Jean-Daniel Boissonnat, Clément Maria. The Simplex Tree: An Efficient Data Structure for General Simplicial Complexes. [Research Report] RR-7993, 2012, pp.20. hal-00707901v1

The idea of representing cochains as skew-symmetric tensors and using them to write the discrete Hodge decomposition was based on the paper:

• Jiang, X., Lim, L., Yao, Y. et al. Statistical ranking and combinatorial Hodge theory. Math. Program. 127, 203–244 (2011). https://doi.org/10.1007/s10107-010-0419-x