Floquet's Theorem, Topology, and Transport in Two Dimensions

In some two-dimensional systems, electrons have topological properties
which endow them with surprising transport properties.  While nature
provides us with a few such materials, their topology is limited by the materials
which can actually be synthesized.  In this talk I will review recent work
in which such topology is induced by time-dependent potentials, allowing
in principle a broad set of possibilities for topological bands to be
created. These "Floquet Topological Insulators" support
surprising fundamental behaviors,  including a quantized Hall effect with
no magnetic field, and in some cases transport enhancement by
disorder. In this talk I will discuss how these possibilities play out for electrons in graphene,
showing how a time-dependent electric field yields a rich set of topological phases,
and how they support phenomena which cannot be realized in a static setting.