  This paper reviews several analytic tools for the field of topological
insulators, developed with the aid of non-commutative calculus and geometry.
The set of tools includes bulk topological invariants defined directly in the
thermodynamic limit and in the presence of disorder, whose robustness is shown
to have non-trivial physical consequences for the bulk states. The set of tools
also includes a general relation between the current of an observable and its
edge index, relation that can be used to investigate the robustness of the edge
states against disorder. The paper focuses on the motivations behind creating
such tools and on how to use them.
