Recently methods of quantum statistical mechanics have been fruitfully applied to the study of phases of quantum lattice systems with interacting gapped Hamiltonians at zero temperature. I will discuss some of these developments focusing on the class of systems known as “invertible”. After introducing the corresponding class of quantum states, I will show how to define H^{d+1}(G,U(1)) valued indices for equivalence classes of such states in the presence of a symmetry group G for d=1 and d=2 lattices. If time permits, I will also discuss invariant for families of such states known as higher Berry curvatures.

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