  One 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 $\lambda$.
For small $\lambda$ and temperature Bose-Einstein condensation (BEC) is proved
to occur, while at large $\lambda$ 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.
