Higher Order Fourier Analysis


Summer school

Alfréd Rényi Institute of Mathematics, Budapest

June 2-4, 2011


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).




June 2, Thursday: 10.00 – 10.50 lecture, 11.05 – 11.55 lecture

June 3, Friday: 10.00 – 10.50 lecture, 11.05 – 11.55 lecture, 16.00 – 18.00 discussion session

June 4, Saturday: 10.00 – 10.50 lecture, 11.05 – 11.55 lecture, 14.00 – 16.00 discussion session


Coffee and tea will be available between the lectures. In the first lecture (Thursday 10.00 – 10.50) the speaker will present a broad overview of the subject.

Remark. The school has an overlap of dates with the Paul Turan Memorial Lectures given by Yuval Peres (june 1 – 3), but there will be no conflict of program.



B. Szegedy: "Gowers norms, regularization and limits of functions on abelian groups" http://arxiv.org/abs/1010.6211
B. Szegedy: "Nilspaces, nilmanifolds and their morphisms" http://arxiv.org/abs/1009.3825
B. Szegedy: "Higher order Fourier analysis as an algebraic theory II." http://arxiv.org/abs/0911.1157
B. Host, B. Kra: "Parallelepipeds, nilpotent groups and Gowers norms" http://www.math.northwestern.edu/~kra/papers/nil.pdf
B. Green, T. Tao, T. Ziegler: "An inverse theorem for the Gowers U^{s+1}(N)-norm" http://arxiv.org/abs/1009.3998
T. Gowers: "A new proof of Szemeredi's theorem" http://www.cs.umd.edu/~gasarch/vdw/sz-thm-gowers-proof.pdf



Here is a short text that was distributed among the participants.