Our appreciation of insulating states of quantum matter has been deepened in the last decade by the theoretical prediction and subsequent experimental discovery of topological insulators. Much of the theoretical discussion of these phases is informed by free electron band theory. In this talk I will describe recent progress in generalizing the concept of topological insulation to strongly interacting electronic systems.

I focus on a minimal generalization known as symmetry-protected topological (SPT) phases. Just like the familiar topological insulators these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. I will show how interactions enable many SPT phases for spin-orbit coupled three dimensional electronic insulators that have no analog in band theory. I describe their physical properties and experimental fingerprints. More generally studies of such SPT phases represent possibly the simplest context for the interplay between interactions, symmetry, and topology in states of quantum matter. I will describe how the insights obtained provide fresh viewpoints on a number of other frontier theoretical problems in quantum condensed matter physics.