Combinatorial number theory and applications

最近在各个领域、环和组中围绕“和-积”和“产品现象”的研究结果导致了从计算机科学到表示理论等大量问题的进展。该提案的目的是进一步探索这些应用背后的几种类型的问题，特别是矩阵空间中的“乘积现象”和有限域上代数函数的“展开性质”。提案中的问题主要是继续进行一系列研究（特别是PI的研究），这些研究结果除了从组合的角度来看它们的内在利益之外，还具有相当意想不到的应用。事实上，纯粹的组合（基本上是基本的）技术在停滞了很长一段时间的问题上取得了进展，比如某些“薄”SL2 Caley图的展开性质，以及有限域上代数函数的展开性质。

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.