Solving nonlinear equations in arithmetic sets.

这是一个关于加性数论的课题。这个数论的子领域广泛涉及整数子集及其在加法作用下的行为的研究。它与素数理论、组合数论和数字几何有着密切的联系。该领域的两个经典问题是哥德巴赫猜想（即所有大于2的偶数都是两个素数的和的猜想）和沃林问题（即对于给定的整数k，是否每个整数都可以表示为有限数量的k次幂的和）。许多这类问题是用Hardy-Littlewood圆法和筛法的工具来研究的。通常，质数集是特别有趣的，这些技术已经成功地应用于几个相关的问题。例如，维诺格拉多夫用它们证明了每一个足够大的奇数都是三个素数的和。本着类似的精神，本项目将以素数计算某些高次方程的解为目标，但由于非线性行为，人们必须超越上述经典方法。因此，该项目具有与其他领域的接口，例如表示理论和组合学。这是一个纯数学领域的项目，目前还没有设想对该学科以外的领域产生影响。该项目的最初目的是更仔细地研究我的导师本·格林最近发表的一篇论文的几个方面。本文利用群论和表示论的方法确定了8变量的一般二次方程解的渐近个数，其中要求变量为素数。各种各样的自然问题出现了。例如，这些方法是否可以适用于除素数以外的整数集，甚至可能适用于本质上任意的充分密集集？这些方法是否可以扩展到处理比齐次二次方程更一般的方程，例如允许线性项？这种方法是否需要当前二次型的通用性假设，或者可以通过进一步的工作来消除它？该项目的后续目标将是更广泛地研究将有限群的表示理论应用于可加数论问题的潜力，或研究大于2次的一般方程。Green的论文是最近发表的（2021年8月），论文中的方法非常新颖。实际上，这篇论文似乎是有限群的表示理论在圆方法中的第一个应用。因此，这些方法扩展了研究数论中可加性问题的传统工具，值得研究它们在可加性数论问题中的普遍适用性。该项目属于EPSRC数论研究领域。

This is a project in additive number theory. This subfield of number theory broadly concerns the study of subsets of integers and their behaviour under addition. It has close ties to prime number theory, combinatorial number theory and the geometry of numbers. Two classical problems in the field are the Goldbach conjecture (which is the conjecture that every even number greater than two is a sum of two primes) and Waring's problem (which asks whether, for a given integer k, every integer can be expressed as a sum of a bounded number of k-th powers). Many of these problems are studied using the tools from the Hardy-Littlewood circle method and from sieve methods. Often the set of prime numbers is of particular interest and these techniques have been applied successfully to several related problems. For example, Vinogradov used them to prove that every sufficiently large odd number is the sum of three primes. In a similar spirit, this project will aim to count solutions to certain higher-degree equations in the prime numbers, but due to the non-linear behaviour one has to go beyond the classical methods mentioned above. Hence, the project has interfaces with other areas such as representation theory and combinatorics. It is a project in pure mathematics and no impact outside of the subject is currently envisaged.The initial aim of the project is to look more closely at several aspects of a very recent paper by my supervisor Ben Green. In this paper, methods of group theory and representation theory are used to determine the asymptotic number of solutions to generic quadratic equations in 8 variables, where the variables are required to be prime. Various natural questions present themselves. For instance, can these methods be adapted to sets of integers other than the primes, perhaps even to essentially arbitrary sufficiently dense sets? Can the methods be extended to handle more general equations than homogeneous quadratics, for example by allowing linear terms? Does this approach require the current genericity assumption on the quadratic form, or can it be removed with further work? Subsequent aims of the project would be to look in greater generality at the potential for applying the representation theory of finite groups to questions in additive number theory, or to look at generic equations of degree greater than 2. The paper of Green is very recent (August 2021) and the methodology in that paper is quite novel. Indeed, the paper appears to be the first application of the representation theory of finite groups to the circle method. Hence, these methods extend traditional tools for studying additive questions in number theory, and it is worth investigating their general applicability to additive number theory problems.This project falls within the EPSRC Number Theory research area.