Abstract: Many believe, following Anderson, that the origin of the unusual behavior of cuprate superconductors is related to the physics of the undoped phase. However, this connection has not been established theoretically. I will discuss the properties of a renormalized Hamiltonian derived recently from the t-J model, which is required to reproduce the physics at half filling. Despite this constraint, or, perhaps because of it, the resulting phases agree qualitatively with those found in underdoped cuprates. I will discuss the origin of strong-coupling superconductivity in the model. One property that is central to the physics of cuprates is the two-dimensional nature of the normal state, although the superconductor and the other nearby phases show the expected 3D behavior. Remarkably, this "dimensional confinement" of the metal emerges in this theory, not put in by hand.