Topological quantum numbers in quasicrystals

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.