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).

SCHEDULE OF LECTURES AND DISCUSSION
SESSIONS.

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.

SOME LITERATURE FOR THE LECTURES:

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.

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/~

A SHORT DICTIONARY

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