Part 1: Geometry, background and basic results

Recently new materials have been synthesized which have the geometry of a double gyroid on the nanometer scale. In particular these material give rise to a wire system. When discretizing this wire system,  we arrive at a specific non-commutative system/space. We will discuss how this system is related to the non-commutative torus via a Harper Hamiltonian. We also treat simpler cases such as the square lattice which leads to the Hofstadter butterfly and the honeycomb lattice which is the underlying structure for graphene. Some of the techniques are from (non)-commutative geometry and K-theory.

This is joint work with Ralph Kaufmann and Sergei Khlebnikov.