Abstract: | We explain the mathematical background for the construction of topological quantum numbers in aperiodic crystals. Often topological quantum numbers are referred to as Chern numbers, as they arose first as Chern numbers of vector bundles defined by solutions of the Schrödinger equation for periodic crystals. We explain how to define them in a non-commutative way which is also applicable to aperiodic crystals. This formulation allows to obtain equations between topological numbers of different physical systems. We present two applications: a particular manifestation of the bulk boundary correspondence which relates the labeling of gaps in the spectrum of a one-dimensional quasiperiodic Hamiltonian to the phason motion and a relation between Bragg peaks in the diffraction spectrum and chern numbers for topological insulators. |