What makes one phase of matter different from another? For much of physics, the answer has been guided by symmetry and its breaking. Yet we now know that this is only part of the story. Topological phases of matter defy this framework: they can have no local order parameter and yet possess robust quantization, protected boundary modes, and long-range entanglement. Their defining features turn out to be encoded not in symmetry alone but in the geometry and topology of the quantum state space — reflected in quantities such as the Berry phase and the quantum metric. Even more surprising developments have emerged in driven systems, which can host dynamical topological phases with properties that have no analog in equilibrium. Among these is the counterintuitive possibility that disorder need not localize — that dynamics can protect extended states even in regimes where Anderson localization would normally prevail. In this talk I will trace this evolution from geometric phases to dynamical topology, emphasizing unifying ideas and open questions. No prior familiarity with topology will be assumed.
Rahul Roy obtained his Ph.D from the University of Illinois under the supervision of Michael Stone in 2007. He went on to do a couple of post-doctoral fellowships: at McMaster University with Catherine Kallin and John Berlinsky and at Oxford University with Steve Simon and John Chalker. He is the recipient of the McMillan Prize and the Sloan Fellowship. His work has primarily focused on topological aspects of condensed matter systems.