  We study topological insulators, regarded as physical systems giving rise to
topological invariants determined by symmetries both linear and anti-linear.
Our perspective is that of noncommutative index theory of operator algebras. In
particular we formulate the index problems using Kasparov theory, both complex
and real. We show that the periodic table of topological insulators and
superconductors can be realised as a real or complex index pairing of a
Kasparov module capturing internal symmetries of the Hamiltonian with a
spectral triple encoding the geometry of the sample's (possibly noncommutative)
Brillouin zone.
