Spatial restriction of exponential sums to thin sets and beyond

近年来，PI开发了一种称为解耦的新工具，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为了分析描述波相消的微分方程，但它也导致了数论的重要突破。例如，丢番图方程是涉及整数的潜在复杂方程组。它们用于生成加扰密钥和扩散密钥，这些密钥在加密数据时非常有用。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。 通过这种方式，与计算丢番图系统的解的数量有关的问题可以用量化波干扰的语言来重新表述。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计划进一步扩展解耦的范围，以解决调和分析、几何测度理论和数论中的基本问题。 他将寻求使新的工具访问和有用的很大一部分数学界。本计画提供研究训练机会给研究生，部份计画的目的在于发展分析薛定谔极大函数在周期性环境下的方法。基于他最近的进展，PI的目标是将傅立叶分析和更精细的数论纳入现有的组合框架。解耦已被证明是成功的，在解决广泛的问题，如数论，偏微分方程和谐波分析等不同领域。目前的项目力求在新的方向上进一步扩大这一方法的适用性。其中之一是关于找到尖锐的估计支持曲流形上的分形测度的傅立叶变换。PI试图将联合收割机解耦与球和管之间的发生率的精确估计相结合。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。这个奖项反映了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。

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.