(Non)--commutative geometry of wire network graphs from triply periodic CMC surfaces

We discuss the classical and non-commutative geometry of wire systems which are the complement of triply periodic surfaces. We consider a C∗-geometry that models their electronic properties. In the presence of an ambient magnetic field, the relevant algebras become non-commutative and basic information can be derived from this description. For the commutative case, which is interesting in its own right, we have developed further tools and methods, such as a re-gauging groupoid, applications of singularity theory and characteristic classes. Our main example is the gyroid, which can be fabricated on the nanoscale. We also compare the gyroid wire network to networks from other CMC surfaces. In this setting the gyroid geometry can be seen as the 3d generalization of graphene.
This is joint work with R. Kaufmann and S. Khlebnikov.