Conventional Fourier analysis has proved useful in counting
three-term arithmetic progressions. This series of talks covers the
recent work of Green and Tao, using *quadratic* Fourier
analysis to count 4-term AP's in subsets of F_5^n. A key tool is applying the following
transformation to the characteristic function of the set:
\[\int_{x\in F_5^n} f(x)\omega^{ x^T Mx+r^T x}\] where \omega is a fifth root of unity, r an element of F^n_5,
(viewed as a vector) and M an n\times n matrix in F^5.

Brooke Orosz