The Non-Commutative Geometry of the Quantum Hall Effect
  We give an overview of the Integer Quantum Hall Effect. We propose a mathematical framework using Non-Commutative Geometry as defined by A. Connes. Within this framework, it is proved that the Hall conductivity is quantized and that plateaux occur when the Fermi energy varies in a region of localized states. 
J. Bellissard, A. van Elst, H. Schulz-Baldes