The coarse index of J. Roe beautifully expands index theory to noncompact manifolds.
It appears abstract at first glance, but I will explain how the coarse index counts:
(1) homological obstruction to quantum localization;
(2) connectedness of spectrum and anomalous currents;
(3) noncommutative families index and stability of Landau quantization.
These perspectives come directly from physics, and motivate NCG for Roe-type algebras.
Includes joint work with M. Ludewig and Y. Kubota.