Combinatorial number theory and applications

最近在不同领域、环和群中围绕“和积”和“积现象”的结果导致了从计算机科学到表示理论的许多惊人的问题上的进展。该建议的目的是进一步探索这些应用的几种类型的问题，特别是矩阵空间中的“乘积现象”和有限域上的代数函数的“扩展性质”。提案中的问题主要是继续一系列研究(特别是由PI进行的)，这些研究被证明具有相当意想不到的应用，除了从组合的角度来看它们的内在兴趣之外。事实上，纯粹的组合(和基本的基础)技术在一些已经停滞了很长一段时间的问题上取得了进展，例如在某些“薄”的SL2 Caley图的扩张器性质上，以及在有限域上的代数函数的“展开性质”上。

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.