Topological Quantum Matter

Nobel Lecture, presented December 8, 2016, Aula Magna, Stockholm University. I will describe the history and background of three discoveries cited in this Nobel Prize: e "TKNN" topological formula for the integer quantum Hall eect found by David ouless and collaborators, the Chern Insulator or quantum anomalous Hall eect, and its role in the later discovery of time-reversal-invariant topological insulators, and the unexpected topological spin-liquid state of the spin-1 quantum antiferromagnetic chain, which provided an initial example of topological quantum matter. I will summarize how these early beginnings have led to the exciting, and currently extremely active, feld of "topological matter."