Topological Calculus: between Algebraic Topology and Electromagnetic fields

The Topological Calculus, based on Algebraic Topology, is introduced as a discrete Field Theory. Diagonalization of simplicial complex adjacency matrices allows to extract information about domain topology and Helmholtz equation eigenfunctions. Electromagnetic analysis of IFS fractals for Sierpinski gasket/carpet is then carried out: self-similar topology deeply influences the type of e.m. fields, as well as its finite TEM modes (as many as the domain's Euler characteristic; represented by harmonic fields) and self-similar distribution of resonating frequencies. This proves that even in such discrete model many features of guided waves depend on the topology rather than metrics.