Topological materials, a form of physical matter with unusual but useful properties, have brought with them unexpected new connections between physics and pure mathematics. As their name suggests, topology has played a significant role in understanding and classifying these materials. In this talk, I will offer a brief look at a vast extension to this story, arising from my work with a number of collaborators over the last five years. This work sees complex geometry — in particular, Riemann surfaces and various moduli spaces arising as Hamiltonian quotients — being used to anticipate new models of quantum matter. Most importantly, calculations may be made regarding the physics of such materials via a hyperbolic realization of Bloch-Floquet band theory, which has also appeared in our work. There will be lots of pictures.