  The first reliable analytic calculation of the phase diagram of the bose gas
on a $d$-dimensional lattice with on-site repulsion is presented. In one
dimension, the analytic calculation is in excellent agreement with the
numerical Monte Carlo results. In higher dimensions, the deviations from the
Monte Carlo calculations are larger, but the correct shape of the Mott
insulator lobes is still obtained. Explicit expressions for the energy of the
Mott and the ``defect'' phase are given in a strong-coupling expansion.
