  This is the second part of the notes to the course on quantum theory of large
systems of non-relativistic matter taught by J. Fr\"{o}hlich at the 1994 Les
Houches summer school. It is devoted to a sketchy exposition of some of the
beautiful and important, recent results of J.Feldman and E.Trubowitz, and J.
Feldman, H. Kn\"{o}rrer, D. Lehmann, J. Magnen, V. Rivasseau and E. Trubowitz.
Their results are about a mathematical analysis of non-relativistic many-body
theory, in particular of the Landau-Fermi liquid and BCS superconductivity,
using Wilson's renormalization group methods and the techniques of the
$1/N$-expansion. While their work is ultimately aimed at a complete
mathematical control (beyond perturbative expansions) of systems of weakly
coupled electron gases at positive density and small or zero temperature, we
can only illustrate some of their ideas within the context of perturbative
solutions of Wilson-type renormalization group flow equations (we calculate
leading-order terms in a $1/N$-expansion, where $N$ is an energy scale) and of
one-loop effective potential calculations of the BCS superconducting ground
state. Contents:
  1. Background material
  2. Weakly coupled electron gases
  3. The renormalization group flow
  4. Spontaneous breaking of gauge invariance, and superconductivity
