Explore cutting-edge concepts in topology through this lecture on "New Frontiers in Topology: Multilevel Systems and Out-of-Equilibrium" delivered by F Nur Ünal. Delve into advanced topics related to periodically and quasi-periodically driven complex systems, focusing on multilevel systems and out-of-equilibrium phenomena. Gain insights into the intersection of closed quantum systems with time-dependent Hamiltonians, open quantum systems, and gauge theory of coupling between matter and radiation. Discover how these fields contribute to understanding terahertz and infrared driven quantum matter, shaken optical lattices, Floquet time crystals, and localization-delocalization physics in quasi-periodically driven systems. Learn about the latest developments in the dynamics of gauge theories and their applications in quantum simulation experiments.