Montreal Lecture Notes on Quadratic Fourier Analysis

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 Some of the notes cover unpublished work which is joint with Terence Tao.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.