Theoretical results for the Gyroid wire system*

We discuss the classical and non-commutative geometry of wire systems which are

the complements of triply periodic surfaces. Our predominant example is the

gyroid surface and its complementary wire network which can be fabricated on a

nanoscale, but we also worked on wire networks stemming from other CMC

surfaces. For all these geometries, we study the Harper Hamiltonian and its band

structure in the commutative and non-commutative cases.  In the latter, an

ambient magnetic field destroys the commutativity of the relevant algebras. We

have successfully applied methods from singularity theory, representation theory

and topology to these materials. In this setting the gyroid geometry can be seen as

the 3d generalization of graphene. We will discuss the most important results of our work.