Some problems in harmonic analysis

首席研究员（PI）打算深入研究位于谐波分析，数论和色散方程交界处的挑战。除了关注经典的傅立叶分析，PI的目标是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，PI计划指导学生，通过谈话传播他们的发现，并促进合作，从而产生更广泛的影响。PI计划继续在几个领域进行研究工作。首先，PI和他的合作者将深入研究快速发展的现代数学领域，特别是专注于傅里叶分析的加性组合。在这个领域内，他们的目标是进一步探索罗斯定理，这是一个基本的结果，决定了在{1，…N}。他们的工作将扩展他们之前对环和/或有限域的多项式罗斯定理的研究。其次，在经典调和分析中，PI致力于研究Sogge和Tao提出的平面上Bochner-Riesz均值的推测的点向收敛性。第三，在与杨的合作中，PI在改进高斯圆问题和狄利克雷除数问题方面取得了长足的进步。他们认为，在这些领域仍有进一步进展的空间。最后，PI将继续他对Waring问题的研究，这个问题可以作为一个函数的解耦问题，其傅里叶变换被限制在一条折线上。该奖项反映了美国国家科学基金会的法定使命，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.