Topological Quantum Matter Research Article • • TQM The Non-Commutative Geometry of the Complex Classes of Topological Insulators

Alain Connes ’ Non-Commutative Geometry program [1] has been recently carried out [2, 3] 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 Ge-ometry involved in these studies. The exposition pro-ceeds gradually through the cyclic cohomology, quan-tized calculus with Fredholm-modules, local formulas for the odd and even Chern characters and index the-orems 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. Keywords topological insulators • non-commutative geometry • aperiodic crystals • A- and AIII-symmetry classes © 2013 Emil Prodan, licensee Versita Sp. z o. o.This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs license (http://creativecommons.org/licenses/by-nc-nd/3.0/),which means that the text may be used for non-commercial purposes, provided creditis given to the author