I will provide analytic background underlying the use of modern topology to discuss phases of matter in physics (Nobel Prize 2016). A "topological" insulator paradoxically turns into a "topological metal" at its boundary, and this defining feature survives a large variety of perturbations. Mathematically, one encounters twisted Schrodinger operators, and discovers that certain spectra are "topologically-protected'', via index theory. The example of quantum Hall systems (Nobel 1985), on general manifolds, will be discussed.