FRG: Collaborative Research: New Trends in Harmonic Analysis

提议的活动将集中于谐波分析中当前感兴趣的一系列主题：Kakeya猜想，傅里叶限制问题和Carleman估计，波动方程的平滑猜想，以及Zygmund关于Lipschitz谱族微分的猜想。同样重要的是正在被识别和研究的新问题，包括经典算子的离散类似物、Fuglede猜想的各个方面、具有退化正则关系的傅立叶积分算子和复分析中奇异积分的新类别。这些学科过去的进展受到组合学、加性组合学和数论等学科的影响，这是一些突出的领域。反过来，这些问题也在为这些领域做出贡献。这项建议所允许的各项努力的协调应能加速在这一系列广泛议题上取得进展。所提议的活动涉及的中心问题将导致新的分析技术模式，这些模式涉及的问题，例如，波在更高维度的行为，以及离散（即数字）和连续物体之间细微区别的不同方面。此外，这些项目将借鉴一系列不同数学领域的方法和技术。该主题分析方法的广度和复杂性为这些领域之间的相互关系提供了新的亮点。它也可能成为继续研究的障碍。这个项目的重点是培养这些新兴研究领域的研究生和博士后，以及在他们的分析中使用的各种各样的技术。这些努力将培养下一代数学家，他们对国家的科学基础设施至关重要。

Abstract for 0456538 Lacey, 0456306 Iosevich, and 0456490 MagyarThe proposed activity will focus a set of topics of current interest in Harmonic Analysis: the Kakeya conjecture, the Fourier restriction problem and Carleman estimates, the smoothing conjecture for the wave equation, and Zygmund's conjecture on the differentiation along Lipschitz families of lines. Equally central are newer questions that are being identified and studied, including the discrete analogues of classical operators, aspects of the Fuglede conjecture, Fourier integral operators with degenerate canonical relations and new classes of singular integrals in complex analysis. Past advances in these subjects have drawn influences from subjects such as Combinatorics, Additive Combinatorics, and Number Theory to name some prominent areas. In turn, these questions are making contributions to these same areas. The coordination of the efforts that this proposal will permit should accelerate advances on this broad range of topics. The proposed activities concern central questions that will result in new modes of analytical technique that bear on questions of, for instance, behavior of waves in higher dimensions, and different aspects of the subtle distinctions between discrete, i.e. digital, and continuous objects. In addition, the projects will draw upon methods and techniques from a range of different areas of mathematics. The breadth and sophistication of the analytical methods in the subject sheds new light on the interrelationships between these areas. It also can be an obstacle to continued research. This project has as an important of focus the training of graduate students and postdocs in these emerging areas of research, and the wide variety of techniques used in their analysis. These efforts will foster a next generation of mathematicians, critical to the nations scientific infrastructure.