  These lectures are centered around a specific problem, the effect of weak
repulsive interactions on the transition temperature $T_c$ of a Bose gas. This
problem provides indeed a beautiful illustration of many of the techniques
which have been discussed at this school on effective theories and
renormalization group. Effective theories are used first in order to obtain a
simple hamiltonian describing the atomic interactions: because the typical
atomic interaction potentials are short range, and the systems that we consider
are dilute, these potentials can be replaced by a contact interaction whose
strength is determined by the s-wave scattering length. Effective theories are
used next in order to obtain a simple formula for the shift in $T_c$: one
exploits there the fact that near $T_c$ the physics is dominated by low
momentum modes whose dynamics is most economically described in terms of
classical fields; the ingredients needed to calculate the shift of $T_c$ can be
obtained from this classical field theory. Finally the renormalization group is
used both to obtain a qualitative understanding, and also as a non perturbative
tool to evaluate quantitatively the shift in $T_c$.
