The failure of band theory became apparent in the 1940's when it was realized that because of strong interactions, half-filled-band materials that should have been metallic were instead insulators, so-called Mott insulators.
The failure of BSC theory became apparent in the 1970's when superconductivity was discovered in the presence of strong repulsion instead of weak phonon-mediated attraction.
In this talk, I will present the results of Cluster Dynamical Mean-Field Theory calculations for the simplest model that embodies the physics of strong interactions, the one-band Hubbard model. The resulting phase diagram shows that the effect of strong-interactions, or of Mott physics if you want, extends far from half-filling. In particular, the phase diagram contains a first-order transition in the normal state at finite doping as well as d-wave superconductivity. The first-order transition separates a pseudogap phase from a correlated metallic state. The pseudogap does not break symmetry and does not come from precursor Cooper pairs. The pseudogap temperature follows the so-called Widom-line of the first order transition, a concept that I will explain. I will show that the phenomenology of the pseudogap and of superconductivity found for strong interactions is very close to that of hole-doped high-temperature superconductors.