A local view on topology

In this talk, I revisit topology in solid-state systems from a local perspective. This view highlights qualitative differences in the physical response to local perturbations between topological and trivial systems, from topological semimetals to insulators. We will discuss the role of crystalline symmetry and concretize it by analyzing the effect of local impurities in these different systems. I will show numerical results for electron localization in the presence of chemical potential disorder, where we find rich and unexpected behavior beyond the conventional theory of electron localization.