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.
This package can be installed using the Julia Package Manager. Simply open the REPL, enter
] and run
pkg> add Hodge
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 on 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 taken from 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