10/6/2005

Terence Tao
UCLA

Quadratic Fourier Analysis

Traditional (linear) Fourier analysis is based on analyzing a function via its correlation with linear phases such as $\exp(2 \pi i \xi x)$. > More recently, some progress has been made in developing quadratic > Fourier analysis, in which one uses correlations with quadratic phases also > to analyze the behaviour of a function. Most notably, this approach has > been very useful for understanding the occurence of progressions of length > four inside a given set (linear Fourier analysis can only handle progressions of > length three). We discuss the state of the art and some recent results in > this area.