2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169543One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In \cite{ALSSY} a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modeled by a periodic potential of strength $λ$. For small $λ$ and temperature Bose-Einstein condensation (BEC) is proved to occur, while at large $λ$ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect. This contribution gives a pedagogical survey of these results.To appear in the proceedings of QMath9, Giens, France, Sept. 12--16, 2004Statistical MechanicsOther Condensed MatterMathematical Physicstext