Motivated by recent theoretical and experimental developments in
the physics of hyperbolic crystals, I will introduce the noncommutative
Bloch transform which I will call the hyperbolic Bloch transform (HBT). The
HBT transforms wave functions on the hyperbolic plane to sections of
irreducible, flat, Hermitian vector bundles over the orbit space and
transforms the hyperbolic Laplacian into the covariant Laplacian. I will
prove that the HBT is injective and “asymptotically unitary”. If time
permits, I will talk about potential applications to hyperbolic band
theory. This is a joint work with Steve Rayan (Annales Henri Poincaré,
2023).