Predicted by Einstein in 1924, the Bose-Einstein condensation is a
striking phase transition that takes place in certain systems of
quantum bosonic particles. The dependence of the critical
temperature on the interparticle interactions has been a controversial
issue in the physics community. I will review the mathematical setting
and the literature, and I will describe rigorous upper bounds for the
critical temperature of dilute systems. This upper bound is expected
to be sharp in 2D but not in 3D. (This is joint work with R. Seiringer.)