Topological invariants and topological insulators

Since their characterization over 80 years ago, physicists have believed that there is only one type of band insulator -- a filled valence band below the Fermi energy with a gap to excitations in the conduction band above the Fermi energy. In the past few years, it has become clear that this is not the whole story: band insulators have topological invariants that distinguish them from each other, and phase transitions must separate insulators with different values of these invariants. Insulators with non-trivial values of these invariants have come to be known as "topological insulators." In this talk I will give simple physical descriptions of these invariants, discuss their implications, and examine the experiments that have actually observed topological insulators. Application of these topological ideas has more recently led to a similar classification of topological superconductors and superfluids, some of which are well known, and others of which have yet to be observed. Finally, I will discuss further applications of topological invariants, as well as current and future directions.