Some problems in harmonic analysis

主要研究者（PI）打算深入研究位于调和分析，数论和色散方程交界处的挑战。除了专注于经典的傅立叶分析，PI旨在建立与不同领域的联系，包括数论，组合学，环面上的色散方程和遍历理论。此外，研究所计划指导学生，通过讲座传播他们的研究成果，并促进合作，从而产生更广泛的影响。研究所计划继续在几个领域开展研究工作。首先，PI和他的合作者将深入研究现代数学的快速发展领域，特别是专注于添加剂组合数学以及傅立叶分析。在这一领域，他们的目标是进一步探索罗斯定理，这是一个基本结果，它确定了{1，.，}内算术级数存在所需的最小子集大小。N}。他们的工作将扩大他们以前的调查多项式罗斯定理环和/或有限域。其次，在经典调和分析中，PI致力于研究平面上Bochner-Riesz平均的约束逐点收敛性，如Sogge和Tao所提出的。第三，与杨合作，PI在改进高斯圆问题和狄利克雷除数问题方面取得了进展。他们认为，在这些领域仍有取得更多进展的余地。最后，PI将继续他对Waring问题的研究，该问题可以作为傅里叶变换被限制在一条折线上的函数的解耦问题来处理。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The principal investigator (PI) intends to delve into challenges situated at the junction of harmonic analysis, number theory, and dispersive equations. In addition to focusing on classical Fourier analysis, the PI aims to establish connections with diverse fields, including number theory, combinatorics, dispersive equations on tori, and Ergodic theory. Furthermore, the PI plans to mentor students, disseminate their findings through talks, and foster collaborations, thereby generating broader impacts.The PI plans to continue the research efforts in several areas. Firstly, the PI and his collaborators will delve into the rapidly advancing field of modern mathematics, particularly focusing on additive combinatorics alongside Fourier analysis. Within this realm, they aim to further explore Roth's theorem, a fundamental result that determines the minimum subset size required for the existence of arithmetic progressions within {1, ..., N}. Their work will extend their previous investigations into the polynomial Roth theorem on rings and/or finite fields. Secondly, in classical harmonic analysis, the PI is dedicated to investigating the conjectured pointwise convergence of the Bochner-Riesz mean on the plane, as proposed by Sogge and Tao. Thirdly, in collaboration with Yang, the PI has made strides in improving both Gauss's circle problem and Dirichlet's divisor problem. They believe there is still room for additional progress in these areas. Finally, the PI will continue his study of the Waring problem, which can be approached as a decoupling problem for a function whose Fourier transform is confined to a broken line.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.