Gapped ground state phases of quantum lattice systems

A salient feature of topological insulators and symmetry protected topological phases is the spectral gap above the ground state. I will review recent results of two kinds. The first kind addresses the existence of a gap for specific systems. The second type of results is concerned with the notion of gapped phase and addresses the question of stability of the spectral gap and other fundamental characteristics of the system under uniformly small perturbations of the interactions in the model Hamiltonian. I will outline basic strategies used to approach both types of problems.