Topics in the microscopic theory of Bose-Einstein condensates

We first review the microscopic characterization of Bose-Einstein condensation. We then discusses possibilities of fragmented condensates, where the single-particle density matrix has more than one macroscopic eigenvalue. Afterwards, we turn to several problems in which it is necessary to go beyond the Gross-Pitaevksii approximation to determine the physics of condensates, in particular, the dependence of the transition temperature on strength, and the physics of the regime where the scattering length becomes comparable to or greater than the interparticle spacing.