Past lectures

Abstract

This will be a self contained course on the Renormalization Group with applications to problems in statistical mechanics, self-avoiding random walk and quantum field theory. Starting with finite dimensional Gaussian integrals we will cover standard topics such as (a) Laplace's approximation and the Feynman graphical expansion for "nearly Gaussian" integrals (b) Gaussian measures on function spaces. The Renormalization Group will enter as a generalization of Mehler's formula for the exponential of the Hermite operator. It will lead us to a major improvement in the expansion in (a) which can approximate functional integrals with much greater uniformity. In many cases it is an open problem to prove this and in others the proofs are very difficult. We will largely confine ourselves to the simpler cases where easier proofs are available.