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

本项目主要在谐波分析领域，特别侧重于揭示时频分析与加性组合学、关联几何和偏微分方程（PDE）等领域的深层联系。该项目研究了三个主要主题：1)傅里叶级数的点向收敛问题（接近可积阈值），2)曲率在与主要开放问题相关的几个模型问题中的作用，包括Zygmund微分猜想和三线性Hilbert变换的有界性，以及3)在PDE和流体力学中的应用。点向收敛的第一个主题是傅里叶分析的基础，已有200多年的历史。在此过程中，它在数学分析（例如时频分析）中产生了新的领域，并为小波理论的发展提供了动力和灵感——小波理论如今在工程（图像处理，数据恢复），医学（MRI）和其他领域中有许多实际应用。第二个主题旨在理解玩具曲线模型，该模型旨在通过遍历理论和数论的应用来揭示长期存在的开放问题。最后，第三个主题涉及诸如最大薛定谔算子的全局行为和水波在二维中奇点形成的研究等问题——后者对我们对物理现实的理解有直接的影响。这是一个涉及谐波分析相关问题的多元化项目，在几个相关领域都有应用。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完全解决了这个问题的空洞模型，特别是验证了Konyagin在2006年国际数学家大会上提出的一个猜想。此外，在他研究lacunary模型的过程中，PI在时频分析和加性组合学之间建立了深刻而令人惊讶的联系，他现在正在利用这种联系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——通过研究一些众所周知的困难开放问题从非平坦模型到平坦模型的转换，旨在加深对曲率在调和分析中的作用的理解——PI开发了一个程序，该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非平坦曲线（及其最大变异体）的Carleson算子。第三个主题与色散偏微分方程和流体力学有着密切的联系。流体力学组件是联合项目的一部分，旨在了解二维环境中两种流体之间界面的渐近几何形状，因为界面接近水波情况下的“飞溅”情况。该项目建立在PI与C. Fefferman和A. Ionescu之前的联合工作的基础上，该工作是关于在无旋转假设下局部光滑双流体界面的二维情况下缺乏飞溅奇点形成。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。

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.