Spatial restriction of exponential sums to thin sets and beyond

指数和对稀疏集及以上的空间限制

• 批准号: 2349828
• 负责人: Ciprian Demeter
• 金额: $ 30万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349828&HistoricalAwards=false
• 关键词: Spatial; restriction; exponential; sums; thin

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.

近年来，PI开发了一种称为解耦的新工具，该工具衡量了朝着不同方向相互作用的波浪的程度。尽管该工具最初旨在分析描述波动取消的不同方程式，但它也导致了数字理论的重要突破。例如，二磷剂方程是涉及整数的潜在复杂方程系统。它们用于生成争夺和扩散键，这些键在加密数据中有用。数学家有兴趣计算此类系统的解决方案的数量。与波浪不同，数字不会振荡，至少不是以明显的方式振荡。但是我们可以将数字视为频率，从而将它们与波浪联系起来。通过这种方式，可以用量化波浪相互作用的语言来重新塑造与计算二聚体系统解决方案的数量有关的问题。 PI在Vinogradov的平均值定理中对主要猜想的突破性解决就是这种情况。 PI计划进一步将解耦的范围扩展到谐波分析，几何测量理论和数量理论中基本问题的范围。他将寻求使新工具可用于数学社区的大部分地区。该项目为研究生提供了研究培训机会。该项目的部分旨在开发分析周期环境中Schrödinger最大功能的方法。在他最近的进步的基础上，PI旨在将傅立叶分析和更精致的数字理论纳入现有的组合框架。事实证明，脱钩成功地解决了诸如数字理论，部分微分方程和谐波分析之类的潜水区域中的广泛问题。当前的项目旨在进一步扩展此方法的适用性。其中一位关心寻找对弯曲歧管支持的分形测量的傅立叶变换的尖锐估计。 PI试图将脱钩与球与管之间的发生率的尖锐估计结合在一起。在另一个方向上，他计划进一步调查新引入的紧密脱钩现象。这与数字理论和lambda（P）估计都有深厚的联系。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响标准，我们是否通过评估诚实地认为我们得到了支持。