Arithmetic Combinatorics and Applications to Number Theory

这项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在问题上的进步需要新的见解和方法；近年来在这一方向上的部分成功依赖于代数和组合学界面上的新技术。这些发展的核心是有限域的所谓算术组合学，它最近取得了重大进展，并提出了新的挑战。本研究项目集中于该领域当前研究的几个问题，包括研究变元上点的阶和带约束变量的代数方程的计数解。有限域上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簇上的点阶的问题(例如，与马尔可夫曲面有关的最新发展)。这项工作的部分动机是研究Markoff三元组的强逼近问题，并估计变量以这样或那样的方式限制时方程的解的数目，例如对乘性群的限制。当经典方法不适用时，有限域和剩余环上的一般和积理论可能有用。各种情况下的和积结果本身就很有趣，因为它们导致了解析数论中的新结果，特别是关于高斯和的估计。

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.