Announcement. There will be a Summer School on Higher Order Fourier Analysis in the Alfréd Rényi Institute of Mathematics on June 2-4, 2011.

The School will center on a series of lectures given by Balázs Szegedy (University of Toronto). In his seminal paper Timothy Gowers introduced a sequence of norms U(k) defined for functions on Abelian groups, later named as Gowers norms. He used these norms to give quantitative bounds for Szemeredi's theorem on arithmetic progressions. The behavior of the U(2) norm is closely tied to classical Fourier analysis, but it was unclear what would be the analogous notion for higher norms. The speaker will present a generalization of Fourier analysis (called higher order Fourier analysis) that solves this problem in a meaningful way. Ordinary Fourier analysis deals with homomorphisms of Abelian groups into the circle group; k-th order Fourier analysis deals with morphisms of Abelian groups into algebraic structures called `k-step nilspaces’. These structures are variants of the parallelepiped structures previously introduced by Host and Kra and they are close relatives of nilmanifolds. The new approach presented by Szegedy has two main components: an underlying algebraic theory of nilspaces and a variant of ergodic theory on ultraproduct groups. Using this theory, one can obtain inverse theorems for the U(k) norms on arbitrary Abelian groups, that generalize results by Green, Tao and Ziegler. This approach leads to a limit theory for functions on Abelian groups in the spirit of the recently developed theory of graph limits. 

There is no registration fee for this school. If you plan to attend or have any questions, please contact one of the organizers, Miklós Abért (abert at renyi dot hu) or László Lovász (lovasz at cs dot elte dot hu).