Arithmetic Combinatorics and Applications to Number Theory

该研究项目涉及算术组合学，这是一个具有许多新兴应用的跨学科研究领域。数论和理论计算机科学问题的进展需要新的见解和方法；近年来这个方向的成功部分依赖于代数和组合学交叉领域的新技术。这些发展的核心是所谓的有限域算术组合，它最近取得了重大进展并提出了新的挑战。该研究项目重点关注该领域当前研究引发的几个问题，包括簇上的点阶的研究以及计算具有约束变量的代数方程的解。有限域中的组合问题继续带来许多挑战。该研究项目特别感兴趣的是涉及有限域上簇的点阶的问题（例如与马尔可夫曲面相关的最新进展）。 这项工作的部分动机是研究马尔可夫三元组的强近似问题，并在变量以某种方式（例如乘法群）受到限制时对方程解的数量进行估计。当经典技术不适用时，有限域和留数环中的一般和积理论可能有用。各种设置中的和积结果本身就很有趣，因为它们会带来解析数论的新结果，特别是对高斯和和短字符和的估计。

This research project concerns arithmetic combinatorics, an interdisciplinary field of research with many emerging applications. Progress on questions in number theory and theoretical computer science requires new insights and methods; part of the success in this direction over recent years relies on novel techniques at the interface of algebra and combinatorics. Central to these developments is the so-called arithmetic combinatorics of finite fields, which has undergone significant recent advances and opened new challenges. This research project focuses on several questions motivated by current research in the area, including study of the orders of points on varieties and counting solutions to algebraic equations with constraint variables.Combinatorial problems in finite fields continue to offer many challenges. Of particular interest in this research project are questions involving orders of points on varieties over finite fields (e.g. recent developments related to the Markoff surface). Part of the motivation for the work is to investigate the problem of strong approximation for Markoff triples and to give estimates on the number of solutions of equations when the variables are restricted one way or another, for instance to multiplicative groups. When classical techniques do not apply, general sum product theory in finite fields and residue rings may be useful. Sum-product results in various settings are of interest in their own right as they lead to new results in analytic number theory, in particular, estimates on Gauss sums and short character sums.