Analytic Methods For Diophantine Problems

该奖项为一个与Hardy-Littlewood圆法的理论和应用有关的项目提供资金。研究者从分析的角度，通过指数和的平均值的估计，以及从算术的角度，通过理解位于特殊子簇上的丢番图系统的平凡解所起的作用，寻求对圆法的改进和详细的理解。通过对作者估计光滑Weyl和的分数阶矩的新方法的进一步研究和发展，以及对指数和的低阶矩的详细研究，将考虑解析的观点。算术观点将通过继续研究者对对称对角方程组的非对角解的个数的研究来考虑，该研究使用切片法，也可能通过筛法。后一种观点也将通过对沃恩和伍利最近提出的“准哈代-利特尔伍德”模型的检验来实现。这项研究属于数论的一般数学领域。数论的历史根源在于对整数的研究，它解决了一些问题，比如一个整数被另一个整数整除的问题。它是数学中最古老的分支之一，出于纯粹的美学原因，人们追寻了许多个世纪。然而，在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。

This award provides funding for a project concerned with the theory and application of the Hardy-Littlewood circle method. The investigator is pursuing an improved, detailed understanding of the circle method both from the analytic viewpoint, through estimates for mean values of exponential sums, and from an arithmetic viewpoint, through an understanding of the role played by the trivial solutions of diophantine systems lying on special subvarieties. The analytic viewpoint will be considered in through further investigation and development of the proposer's new method for estimating fractional moments of smooth Weyl sums and through detailed investigations of low moments of exponential sums. The arithmetic viewpoint will be considered by continuing the investigator's research on the number of non-diagonal solutions of systems of symmetric diagonal equations using slicing methods and also perhaps by sieve methods. The latter viewpoint will also be pursued through the testing of the "Quasi- Hardy-Littlewood" model suggested recently by Vaughan and Wooley. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.