My research program is focused on the study of elastic deformation and topological defects in two materials: graphene and topological insulators. In both systems at low energies the electrons have a nearly linear spectrum, i.e. they behave like relativistic fermions. This allows the study of the effects of defects and deformations on the dynamics of electrons trough the formalism of Dirac equation on curved space-time. In this setting it’ s possible to derive correction to observables properties of the systems like the conductivity for example. In the case of graphene I have derived the contribution to conductivity in the Born approximation of the metric arising from the so-called bumps and made a comparison with the scattering on the gauge potential arising from the elastic deformation. A particular defect, the edge dislocation, is found to be a possible responsible for the behaviour of the conductivity at low energies. The topological insulators are a class of band insulators showing gapless edge states, capable of conduction. This situation is similar to Quantum Hall Effect, both physically and formally. Indeed, as in QHE topological invariants (Chern numbers) classify the behaviour of the material. I am thus focused on the study of these material both formally, on the ground of differential geometry, and physically, studying topological defect in topological insulators. Further investigation has been devoted to the analysis of electron-phonon interaction at the surface of a 3D TI, analysing superconductive instability.