  These are notes to accompany four lectures that I gave at the School on
Additive Combinatorics, held in Montreal, Quebec between March 30th and April
5th 2006.
  My aim is to introduce ``quadratic fourier analysis'' in so far as we
understand it at the present time. Specifically, we will describe ``quadratic
objects'' of various types and their relation to additive structures,
particularly four-term arithmetic progressions.
  I will focus on qualitative results, referring the reader to the literature
for the many interesting quantitative questions in this theory. Thus these
lectures have a distinctly ``soft'' flavour in many places.
  Some of the notes cover unpublished work which is joint with Terence Tao.
This will be published more formally at some future juncture.
