Скачать или смотреть Terence Tao: The circle method from the perspective of higher order Fourier analysis

Higher order Fourier analysis is a collection of results and methods that can be used to control multilinear averages (such as counts for the number of four-term progressions in a set) that are out of reach of conventional linear Fourier analysis methods (i.e., out of reach of the circle method). One notable feature of this theory is that the role of linear phase functions is replaced by the notion of a nilsequence. On the other hand, key identities from linear Fourier analysis, such as the Plancherel identity or the Fourier inversion formula, are notably absent in the higher order theory. In this survey talk we give an introduction to the higher order Fourier theory by revisiting the linear circle method from a higher order perspective, in particular downplaying as much as possible the role of Fourier identities.