Finite volume (or area) models for topological insulators are closer to experiment than in_nite volume models. However, they are only indirectly connected to Brillouin zone and so we need to look elsewhere for a topological space. Fortunately, for three or more noncommuting hermitian matrices, there is an associated joint spectrum that can be a rich topological space. Utilizing the K-theory of such a space we can quantify the stability of bound states in_nite models and predict the instability caused by disorder and defect.

Joint spectrum stabilized by K-theory applies to topological insulators in many symmetry classes and many dimensions and even arises in the study of D-branes. This talk, however, will focus on Chern insulators, especially those based on a quasicrystal. In this case it seems that it is only possible to deduce the index of the relevant Fredholm operator using computer calculation on_nite models.