  The theory of ultracold, dilute Bose gases is the subject of intensive
studies, driven by new experimental applications, which also motivate the study
of Bose-Einstein condensation (BEC) in low dimensions. From the theoretical
point of view there are few, quite special, models in which we are able to
prove BEC for interacting bosons. With the aim of studying the condensation
problem I considered a simplified model for a three and two dimensional system
of bosons interacting with a repulsive short range potential, obtained by
introducing an ultraviolet momentum cutoff. I proved that the interacting
theory is well defined at all orders in terms of one (in three dimensions) or
two (in two dimensions) effective parameters, related to the intensity of the
interaction. Moreover the correlations do not exhibit anomalous dimensions,
i.e. the model is in the same universality class of the exactly soluble
Bogoliubov model. The result is obtained by using Rigorous Renormalization
Group techniques and implementing local Ward identities (WI), which reduce the
number of independent running couplings, this fact being crucial for the
control of the two dimensional theory, where the effective interactions are
relevant. Since the ultraviolet momentum cutoff breaks local gauge invariance,
the corrections to the formal WI have been considered.
