Combinatorial number theory and applications

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Recent results around the 'sum-product' and 'product-phenomena' in various fields, rings and groups lead to progress on an amazing number of issues, ranging from computer science to representation theory. The purpose of the proposal is to explore further several types of questions that underlie these applications, in particular the 'product-phenomena' in matrix spaces and the 'expanding properties' of algebraic functions on finite fields. The problems in the proposal are mainly continuing a line of research (in particular by the PI) that turned out to have rather unexpected applications besides their intrinsic interest from the combinatorial point of view. Indeed, purely combinatorial (and basically elementary) techniques brought progress on issues that had stalled for quite some time, such as on the expander properties of certain 'thin' SL2 Caley graphs, and the 'expanding properties' of algebraic functions on finite fields.