FRG: Collaborative Research: New Trends in Harmonic Analysis

摘要摘要0456538莱西，0456306 Iosevich和0456490 Magyar的摘要拟议的活动将集中在调和分析的当前感兴趣的一组主题：Kakeya猜想，傅立叶限制问题和Carleman估计，平滑猜想的波动方程，和Zygmund的猜想上的微分沿着Lipschitz家庭的线。同样重要的是新的问题，正在确定和研究，包括离散类似的经典运营商，方面的Fuglede猜想，傅立叶积分运营商与退化的典型关系和新的类奇异积分在复杂的分析。这些学科过去的进展已经受到组合学、加法组合学和数论等学科的影响。反过来，这些问题也对这些领域做出了贡献。协调这项建议所允许的各种努力，应能加快在这一广泛议题上取得进展。拟议的活动涉及的核心问题，将导致新的模式的分析技术，承担的问题，例如，行为的波在更高的维度，和不同方面的微妙区别之间的离散，即数字，和连续的对象。此外，这些项目将借鉴数学的一系列不同领域的方法和技术。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视角。它也可能成为继续研究的障碍。该项目的一个重要重点是在这些新兴研究领域培训研究生和博士后，以及在分析中使用的各种技术。这些努力将培养下一代数学家，这对国家的科学基础设施至关重要。

ABSTRACTAbstract for 0456538 Lacey, 0456306 Iosevich, and 0456490 Magyar The 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.