I will present results on the dynamics and stability in a strongly-interacting resonantly-driven two-band model. Using exact numerical simulations, a stable regime at large driving frequencies is found where the dynamics is governed by a local Floquet Hamiltonian that is approximately conserved out to very long times. For slow driving, on the other hand, the system becomes unstable and heats up to infinite temperature. While thermalization is quick in these two regimes (but to different ``temperatures''), in the crossover between them we find slow glassy dynamics: temporal fluctuations become strong and temporal correlations long-lived. Microscopically, the origin of this glassy dynamics is traced back to the appearance of rare Floquet many-body resonances, whose proliferation at lower driving frequency removes the metastable energy conservation, and thus produces thermalization to infinite temperature. If time permits, I will briefly discuss the effects of these Floquet many-body resonances on adiabatic state preparation in the presence of a periodic drive. I will also demonstrate how the widely-used inverse-frequency expansion does not capture these many-body resonances, whereas the recently developed generalisation of the Schrieffer-Wolff transformation for periodically-driven systems does so.
