The C*-algebra of sequin fabric

The (co)sphere bundle on Riemannian manifolds can be imagined in 2 dimensions as a kind of sequin fabric, the discs-ornamented material incorporated in many carnival costumes. I'll give a gentle introduction to pseudodifferential operators, and show that their principal symbol lives as a function on the cosphere bundle. Then, I'll discuss how these operators give rise to a flow on the cosphere bundle -- for the Laplace operator this is the geodesic flow. Writing this in a C*-algebraic manner, the cosphere bundle and geodesic flow can be generalised to noncommutative geometry. Based on joint work with Edward McDonald, I will show how this can be connected to the spread of eigenfunctions (Quantum Ergodicity) via spectral truncations of spectral triples -- connecting to work by Connes and van Suijlekom.