Arithmetic Combinatorics and Applications to Number Theory

本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;近年来在这个方向上的成功部分依赖于代数和组合学接口的新技术。这些发展的核心是所谓的有限域算术组合学，它最近取得了重大进展，并提出了新的挑战。本研究课题的研究内容主要集中在簇上点的阶的研究、带约束变量的代数方程的解的计数等几个问题上。有限域上的组合问题仍然面临着许多挑战。在这个研究项目中特别感兴趣的是涉及有限域上各种点的顺序的问题（例如与马尔可夫曲面相关的最新进展）。 部分动机的工作是调查问题的强逼近马尔可夫三元组，并给予估计的数量解决方案的方程时，变量受到限制的一种方式或另一种方式，例如乘法组。当经典技术不适用时，有限域和剩余环中的一般和积理论可能有用。和产品的结果在各种设置是感兴趣的，因为它们导致新的结果在解析数论，特别是估计高斯和短字符和。

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.