From laser-driven quantum Hall matter to shaken quasicrystals

External driving can modify the properties of a static system and create unconventional phases of matter. In particular, the interplay between external modulation and localization can lead to exotic and tunable transport. We discuss recent theoretical and experimental advances in this area using ultracold atoms in quasiperiodic lattices. A 1D driven quasicrystal can be mapped via the Harper-Hofstadter mapping to a 2D electron gas at a high magnetic field illuminated by arbitrarily polarized light. Tuning the polarization of the light which drives this integer quantum Hall system, we experimentally and theoretically reveal a tessellated phase diagram featuring a nested duality-protected pattern of metal-insulator transitions, and generate anomalous critical transport that is neither ballistic nor localized.