This dissertation comprises two parts, each of which details an investigation within topological matter.

In the first part, we investigate the real space localization properties of static topological insulators in two dimensions. We develop three numerical procedures each of which removes an extensive fraction of the local degrees of freedom. We probe the real space properties of the residual projection operator, showing that the maximally localized length scale ξ supported in the residual Hilbert space diverges as the fraction of states remaining approaches 0. The power-law exponent ν = 1/2 characterizing the singular behavior of ξ appears universal across class A insulators with non-zero Chern number.

In the second part, we explore a class of Floquet topological phases in three dimensions. These phases have translational invariance but no other symmetry, and exhibit anomalous transport at a boundary surface. We show that the boundary behavior of such phases falls into equivalence classes up to local 2D unitary evolution. This provides a classification of the 3D bulk, which we conjecture to be complete. We demonstrate that such phases may be generated by exactly solvable “exchange drives” in the bulk. The edge behavior of a general exchange drive in two or three dimensions is shown to derive from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.