Quantum spin Hamiltonians have been used to model a broad variety of phenomena in condensed matter physics since the early years of quantum theory. More recently, they have been employed in the theory of quantum information and computation. Much of what we know about the ground states of quantum spin
Hamiltonians is derived from the exact solutions that can be found for a relatively small number of models. It is essential, however, that we develop our understanding of the generic behaviors of this class of models. In this talk I will discuss recent work aimed at a qualitative theory of gapped ground state phases of quantum spin
systems in one and more dimensions.
