Recently, my research has focused on two areas of quantum condensed matter physics. The first area is the study of "topological phases" of matter, such as quantum Hall liquids and topological insulators. These phases have a rich internal structure, but unlike conventional phases like magnets or superconductors, this structure has nothing to do with symmetry breaking or order parameters. Instead, the defining features of these phases have a topological character. As a result, entirely new concepts and tools need to be constructed to understand these systems. Much of my research is devoted to developing these new methods and approaches. My second area of focus is at the intersection of quantum information theory and condensed matter physics. Here the fundamental problems are (1) to determine which quantum many-body systems can be efficiently simulated on a classical computer and (2) to develop methods to accomplish this task. In addition to its potential practical implications, this problem is closely related to many basic conceptual questions such as the nature of entanglement in many-body ground states and the classification of gapped quantum phases of matter.