CAREER: Decoupling Theory, Oscillatory Integral Theory, and Their Applications in Analytic Number Theory and Combinatorics

这个项目涉及调和分析和解析数论的研究。调和分析研究一般函数如何用简单函数的和来表示，例如三角函数。解析数论是数论的一个分支，它使用数学分析的工具来回答有关整数的问题。最近，谐波分析工具已被证明是非常有效的回答问题源于解析数论。该项目继续在这两个领域的交叉点进行调查，希望在它们之间创建一个字典，使一个领域的研究人员能够将他们的工具翻译到另一个领域。该项目还为本科生提供了进入当前研究前沿的机会。PI计划举办暑期学校，吸引更多本科生和研究生早期学生进行调和分析和解析数论的跨学科研究，项目涉及解耦理论，振荡积分理论及其在解析数论和组合学中的应用。解耦理论发展的一个目标是获得所有多项式曲面的精确解耦不等式，这些曲面是平移-伸缩不变的（简称TDI）。这意味着丢番图方程的所有TDI系统的积分解的数量上有严格的界限。作为实现这一目标的第一步，PI将研究两个特别重要的系统：单项式的TDI系统和由一个多项式生成的TDI系统。在振荡积分理论中，PI也将研究两个问题。第一个问题是求平均算子沿着矩曲线的最优Sobolev正则性估计。第二个问题要求尖锐的勒贝格估计最大平均算子沿着矩曲线。预计线性波动方程和相关解耦不等式的局部平滑估计将在这两个问题的研究中发挥关键作用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project concerns research in harmonic analysis and analytic number theory. Harmonic analysis studies how general functions can be represented by sums of simpler functions, for instance, trigonometric functions. Analytic number theory is a branch of number theory that uses tools from mathematical analysis to answer questions concerning the integers. Recently, tools in harmonic analysis have proven to be very efficient in answering questions originating from analytic number theory. This project continues the investigation in the intersection of these two fields, in the hope of creating a dictionary between them that would allow researchers in one field to translate their tools to the other. The project also provides undergraduate students access to the forefront of current research. The PI plans to organize summer schools to attract more undergraduate and early graduate students to carry out interdisciplinary research in harmonic analysis and analytic number theory.The project involves work in decoupling theory, oscillatory integral theory, and their applications in analytic number theory and combinatorics. One goal in the development of decoupling theory is to obtain sharp decoupling inequalities for all polynomial surfaces that are translation-dilation invariant (TDI in short). This would imply sharp bounds on the number of integral solutions to all TDI systems of Diophantine equations. As a first step towards this goal, the PI will study two particularly important systems: TDI systems of monomials and TDI systems generated by one polynomial. In oscillatory integral theory, the PI also will study two problems. The first problem asks for the optimal Sobolev regularity estimates for an averaging operator along moment curves. The second problem asks for sharp Lebesgue estimates for maximal averaging operators along moment curves. It is expected that local smoothing estimates for linear wave equations and related decoupling inequalities will play key roles in the study of these two problems.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.