The classification of topological insulators is typically realized using topological band theory.  Band theory relies on periodicity so one can take a form of the Fourier transform and work on a reciprocal lattice.  This leads to a picture of the Hamiltonian as a set of small matrices parameterized over a torus.  The resulting parameterized eigenvalues and eigenvectors lead to what physicists call  Bloch band structure and Bloch eigenvectors.  These can then be used to used to construct a topological invariant.

Here I will talk about a complementary approach that uses the spectral localizer in which the position matrices and the Hamiltonian matrix of the system are used to draw a local picture of the topology of the system. First, I will show that the spectral localizer is able to characterize the topology in nonlinear systems where band theory is not applicable. Then I will show the spectral localizer allows us to describe the topology in systems described by full-wave equations via discretization by finite-element methods.