Sieve in Expansion

This report presents recent works extending sieve methods, from their classical setting, to new situations characterized by the targeting of sets with exponential growth, arising often from discrete groups like SLm(Z) or sufficiently big subgroups. A recent lecture of Sarnak [54] mentions some of the original motivation (related to the Markov equation and closed geodesics on the modular surface). The first general results concerning these sieve problems appeared around 2005 in preprint form, and Bourgain, Gamburd and Sarnak have written a basic paper presenting its particular features [4]. Other applications, with a very different geometric flavor, also appeared independently around that time, first (somewhat implicitly) in some works of Rivin [52].