Fourier theory has been a useful analytic tool in studying discrete
structures. Some of the areas where this theory has been particulaly
fruitful are additive combinatorics, eigen values of graphs and random
walks on finite groups or in the study of Boolean functions used in
computer sciences. A few examples of striking results are the Green-Tao Theorem (2008) that asserts that primes contain arbitrarily long arithmetic progressions, Bourgain’s (2002) general bound of the Fourier spectrum of Boolean functions on {0,1}n or the Lubotzky-Phillips-Sarnak (1988) construction of an explicit infinite family of Ramanujan graphs.
The principal aim of this school will be to prepare young researchers to understand developements in this area. The first week would be entirely devoted to preparatory material on discrete fourier analysis, graph theory, analytic and combinatorial number theory and representation theory of groups to familiarise the eventually uninitiated participant. These topics will be be dealt with in more details in the more specialized courses of the second week.