  Alain Connes' Non-Commutative Geometry program [Connes 1994] has been
recently carried out [Prodan, Leung, Bellissard 2013, Prodan, Schulz-Baldes
2014] for the entire A- and AIII-symmetry classes of topological insulators, in
the regime of strong disorder where the insulating gap is completely filled
with dense localized spectrum. This is a short overview of these results, whose
goal is to highlight the methods of Non-Commutative Geometry involved in these
studies. The exposition proceeds gradually through the cyclic cohomology,
quantized calculus with Fredholm-modules, local formulas for the odd and even
Chern characters and index theorems for the odd and even Chern numbers. The
characterization of the A- and AIII-symmetry classes in the presence of strong
disorder and magnetic fields emerges as a natural application of these tools.
