Spatial restriction of exponential sums to thin sets and beyond

近年来，PI开发了一种名为“去耦”的新工具，可以测量不同方向传播的波相互作用的程度。虽然这个工具最初是用来分析描述波消的微分方程的，但它也导致了数论的重要突破。例如，丢番图方程是包含整数的复杂方程组。它们被用来生成置乱和扩散密钥，这是加密数据的工具。数学家对计算这类系统的解的数量很感兴趣。与波不同，数字不会振荡，至少不会以一种明显的方式振荡。但我们可以把数字看作频率，从而把它们与波联系起来。这样，与计算丢番图系统的解的数目有关的问题可以用量化波干涉的语言来重新表述。这就是PI突破解决维诺格拉多夫中值定理中的主要猜想的情况。PI计划进一步扩大解耦的范围，以解决谐波分析、几何测度理论、数论等基本问题。他将努力使这些新工具能够为数学界的大部分人所用。本项目为研究生提供研究训练机会。本计画的部分目的在于发展分析周期设定中Schrödinger最大函数的方法。以他最近的进展为基础，PI的目标是将傅里叶分析和更精细的数论纳入现有的组合框架。解耦在解决数论、偏微分方程和谐波分析等不同领域的广泛问题方面已被证明是成功的。目前的项目旨在进一步扩大该方法在新方向上的适用性。其中之一是关于寻找在弯曲流形上支持的分形测度的傅里叶变换的尖锐估计。PI试图将解耦与球与管之间的发生率的精确估计结合起来。在另一个方向上，他计划进一步研究新引入的紧密脱钩现象。这与数论和Lambda(p)估计都有很深的联系。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。

In recent years, the PI has developed a new tool called decoupling that measures the extent to which waves traveling in different directions interact with each other. While this tool was initially intended to analyze differential equations that describe wave cancellations, it has also led to important breakthroughs in number theory. For example, Diophantine equations are potentially complicated systems of equations involving whole numbers. They are used to generate scrambling and diffusion keys, which are instrumental in encrypting data. Mathematicians are interested in counting the number of solutions to such systems. Unlike waves, numbers do not oscillate, at least not in an obvious manner. But we can think of numbers as frequencies and thus associate them with waves. In this way, problems related to counting the number of solutions to Diophantine systems can be rephrased in the language of quantifying wave interferences. This was the case with PI's breakthrough resolution of the Main Conjecture in Vinogradov's Mean Value Theorem. The PI plans to further extend the scope of decoupling toward the resolution of fundamental problems in harmonic analysis, geometric measure theory, and number theory. He will seek to make the new tools accessible and useful to a large part of the mathematical community. This project provides research training opportunities for graduate students.Part of this project is aimed at developing the methodology to analyze the Schrödinger maximal function in the periodic setting. Building on his recent progress, the PI aims to incorporate Fourier analysis and more delicate number theory into the existing combinatorial framework. Decouplings have proved successful in addressing a wide range of problems in such diverse areas as number theory, partial differential equations, and harmonic analysis. The current project seeks to further expand the applicability of this method in new directions. One of them is concerned with finding sharp estimates for the Fourier transforms of fractal measures supported on curved manifolds. The PI seeks to combine decoupling with sharp estimates for incidences between balls and tubes. In yet another direction, he plans to further investigate the newly introduced tight decoupling phenomenon. This has deep connections to both number theory and the Lambda(p) estimates.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.