A noncommutative approach to topological insulators

Abstract: 

Topological insulators are characterised by quantum mechanical Hamiltonians that obey particular (possibly anti-linear) symmetries, and give rise to edge states `protected’ by these symmetries. We will outline a general framework that links such Hamiltonians and symmetries to operators algebras and K-theory, both real and complex. By using Kasparov’s bivariant KK-theory, we derive concrete expressions for the quantities of interest as well as a mathematical formulation of the so-called bulk-edge correspondence. This is joint work with A. Carey, J. Kellendonk and A. Rennie.