Arithmetic Combinatorics and Applications to Number Theory

这个研究项目涉及算术组合，这是一个跨学科的研究领域，有许多新兴的应用。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，这方面的成功部分依赖于代数和组合学界面上的新技术。这些发展的核心是所谓的有限域的算术组合学，它最近取得了重大进展，并提出了新的挑战。本研究项目主要关注该领域当前研究的几个问题，包括研究变量上点的阶数和具有约束变量的代数方程的计数解。有限域的组合问题继续提供许多挑战。在这个研究项目中特别感兴趣的是涉及有限域上变异点的阶数的问题（例如最近与马尔科夫曲面相关的发展）。这项工作的部分动机是研究马尔科夫三元组的强逼近问题，并在变量以某种方式限制时（例如乘法群）给出方程解的数量估计。当经典技术不适用时，有限域和剩余环的一般和积理论可能是有用的。各种情况下的和积结果本身就很有趣，因为它们会导致解析数论中的新结果，特别是高斯和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.