Topics in Harmonic Analysis: Time-frequency Analysis and connections with Additive Combinatorics and Partial Differential Equations

谐波分析主题：时频分析以及与加性组合和偏微分方程的联系

• 批准号: 1900801
• 负责人: Victor Lie
• 金额: $ 25万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-15 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900801&HistoricalAwards=false
• 关键词: Topics; Harmonic; Analysis; Time; frequency

This project lies mainly within the area of harmonic analysis, with a special focus on revealing the deep connections between time-frequency analysis and fields such as additive combinatorics, incidence geometry, and partial differential equations (PDE). The project investigates three major themes: 1) the problem of the pointwise convergence of Fourier series (near the integrability threshold), 2) the role of curvature in several model problems connected with major open problems including Zygmund's differentiation conjecture and the boundedness of the trilinear Hilbert transform, and 3) applications in PDE and fluid mechanics. The first theme of pointwise convergence lies at the very foundation of Fourier analysis and has a history of more than 200 years. Along the way, it has generated new areas within mathematical analysis (e.g. time-frequency analysis) and served as a motivation and inspiration for the development of wavelet theory -- a field that nowadays has numerous real-world applications in engineering (image processing, data recovery), medicine (MRI), and other fields. The second theme aims to understand toy curved models designed to shed new light on longstanding open problems with applications to ergodic theory and number theory. Finally, the third theme deals with questions such as the global behavior of the maximal Schrodinger operator and the study of singularity formation in two dimensions for water waves -- the latter having direct implications for our understanding of physical reality.This is a diverse project involving relevant problems in harmonic analysis, with applications in several related fields. The PI has obtained relevant progress on all three of the themes discussed above. Indeed, for the first theme, the PI completely solved the lacunary model of the problem, in particular verifying a conjecture posed by Konyagin at the 2006 International Congress of Mathematicians. Moreover, in the course of his work on the lacunary model, the PI established deep and surprising connections between time-frequency analysis and additive combinatorics, which he is now using to develop a new methodology for approaching the longstanding full problem. For the second theme -- meant to develop a deeper understanding of the role of curvature in harmonic analysis, by studying the transition from nonflat to flat models of some difficult well-known open problems -- the PI developed a program which recently unified three directions: the Hilbert transform, the bilinear Hilbert transform, and the Carleson operator along non-flat curves (together with their maximal variants). The third theme has strong connections with dispersive PDE and fluid mechanics. The fluid mechanics component is part of a joint project and aims to understand the asymptotic geometry of the interface between two fluids in a 2D setting as the interface approaches a "splash" scenario from the water-wave case. This project builds on the previous joint work of the PI together with C. Fefferman and A. Ionescu on the lack of splash singularity formation in the 2D case of locally smooth two-fluid interfaces under irrotational assumptions.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.

该项目主要在谐波分析领域，特别关注揭示时频分析与加法组合学，关联几何和偏微分方程（PDE）等领域之间的深层联系。 该项目研究三个主要主题：1）傅立叶级数的逐点收敛问题（接近可积性阈值），2）曲率在与主要开放问题（包括Zygmund微分猜想和三线性希尔伯特变换的有界性）相关的几个模型问题中的作用，以及3）在PDE和流体力学中的应用。第一个问题是点态收敛，它是傅立叶分析的基础，已有200多年的历史。沿着的方式，它产生了新的领域内的数学分析（例如，时间-频率分析），并作为一个动机和灵感的发展小波理论-一个领域，现在有许多现实世界中的应用工程（图像处理，数据恢复），医学（MRI），和其他领域。第二个主题旨在了解玩具弯曲模型，旨在揭示新的光长期开放的问题与应用遍历理论和数论。 最后，第三个主题涉及的问题，如全球行为的最大薛定谔算子和研究的奇异性形成在两个维度的水波-后者有直接影响我们的理解物理现实。这是一个多样化的项目，涉及相关问题的调和分析，在几个相关领域的应用。PI在上述所有三个主题上都取得了相关进展。事实上，对于第一个主题，PI完全解决了该问题的缺陷模型，特别是验证了Konyagin在2006年国际数学家大会上提出的猜想。 此外，在他对空缺模型的研究过程中，PI在时频分析和加法组合学之间建立了深刻而令人惊讶的联系，他现在正在使用这种联系来开发一种新的方法来解决长期存在的完整问题。对于第二个主题--通过研究从非平坦模型到平坦模型的一些困难的知名开放问题，旨在更深入地了解曲率在调和分析中的作用--PI开发了一个程序，最近统一了三个方向：希尔伯特变换，双线性希尔伯特变换和沿沿着非平坦曲线的Carleson算子（以及它们的最大变量）。 第三个主题与色散偏微分方程和流体力学有很强的联系。流体力学部分是一个联合项目的一部分，旨在了解在二维环境中两种流体之间的界面的渐近几何形状，因为界面接近水波情况下的“飞溅”场景。 该项目建立在PI与C. February man和A. Ionescu在非旋转假设下局部光滑的两流体界面的二维情况下缺乏飞溅奇点的形成。该奖项反映了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。