Geometry and Topology in Motion*

Topological and geometric ideas are now a mainstay of condensed matter physics, underlying much of our understanding of conventional materials in terms of defects and geometric frustration in ordered media, and protected edge states in topological insulators. In this talk, I will argue that such an approach successfully identifies the relevant physics in metamaterials and active matter as well, even when traditional techniques fail. Novel topological phenomena acquire unique realizations in active fluids that pose a particular challenge to conventional statistical mechanics, by virtue of being far from equilibrium. I will present some recent results to illustrate this and show how spontaneous flow on complex substrates can mimic synthetic gauge fields allowing them to be fruitfully viewed using topological ideas and analogies with superconductors.