  The search for strong topological phases in generic aperiodic materials and
meta-materials is now vigorously pursued by the condensed matter physics
community. In this work, we first introduce the concept of patterned resonators
as a unifying theoretical framework for topological electronic, photonic,
phononic etc. (aperiodic) systems. We then discuss, in physical terms, the
philosophy behind an operator theoretic analysis used to systematize such
systems. A model calculation of the Hall conductance of a 2-dimensional
amorphous lattice is given, where we present numerical evidence of its
quantization in the mobility gap regime. Motivated by such facts, we then
present the main result of our work, which is the extension of the Chern number
formulas to Hamiltonians associated to lattices without a canonical labeling of
the sites, together with index theorems that assure the quantization and
stability of these Chern numbers in the mobility gap regime. Our results cover
a broad range of applications, in particular, those involving
quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically
generated) lattices.
