Solving nonlinear equations in arithmetic sets.

这是加性数论中的一个项目。数论的这一子领域广泛地关注整数子集及其在加法下的行为的研究。它与素数论、组合数论和数的几何学有着密切的联系。该领域的两个经典问题是哥德巴赫猜想（这是一个猜想，即每个大于2的偶数是两个素数的总和）和华林问题（这是问是否，对于给定的整数k，每个整数可以表示为一个有限的k次幂的总和）。许多这些问题的研究使用的工具，从哈代-利特尔伍德圆的方法和筛的方法。通常素数的集合是特别感兴趣的，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，该项目的目标是计算素数中某些高阶方程的解，但由于非线性行为，必须超越上述经典方法。因此，该项目与其他领域，如表示论和组合学有接口。这是一个纯数学的项目，目前没有任何影响以外的主题设想。该项目的最初目的是更密切地关注几个方面的一个非常最近的文件由我的导师本绿色。本文利用群论和表示论的方法，研究了8元一般二次方程解的渐近数，其中要求变量为素数.各种自然的问题出现了。例如，这些方法是否适用于素数以外的整数集合，甚至适用于基本上任意的足够稠密的集合？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？这种方法是否需要二次型上的通用性假设，或者可以通过进一步的工作删除它？该项目的后续目标将是寻找更大的普遍性的潜力，适用于有限群的表示理论的问题，在添加剂数论，或看看通用方程的程度大于2。绿色的论文是最近的（2021年8月），该论文中的方法相当新颖。事实上，这篇论文似乎是有限群表示论首次应用于圆方法。因此，这些方法扩展了传统的工具来研究数论中的加法问题，值得研究它们对加法数论问题的普遍适用性。

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.