FRG: Collaborative Research: New Trends in Harmonic Analysis

FRG：协作研究：谐波分析的新趋势

• 批准号: 0456538
• 负责人: Michael Lacey
• 金额: --
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456538&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; New; Trends

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.

0456538 LACEY，0456306 IOSEVICH和0456490 MAGYARTHE THE ASSUD活动的摘要将集中在谐波分析中当前兴趣的主题：Kakeya，Kakeya限制性问题和Carleman估计值，Wave方程和Zygmund的平滑猜想，以及Zygmund的平滑猜想。同样的核心是正在识别和研究的较新问题，包括经典运营商的离散类似物，fuglede猜想的各个方面，具有退化规范关系的傅立叶积分运算符以及复杂分析中的新类奇异积分类别。这些主题的过去进步受到了组合学，添加剂组合学和数字理论等主题的影响，以列举一些重要领域。 反过来，这些问题为这些相同领域做出了贡献。 该提案将允许的努力的协调应加速在这一广泛主题上的进步。拟议的活动涉及的主要问题将导致新的分析技术模式，这些技术在较高维度中的波浪行为以及离散（即数字和连续物体）之间微妙区别的不同方面存在问题。 此外，这些项目将利用来自各种数学领域的方法和技术。 对象中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启示。 这也可能是继续研究的障碍。 该项目非常重要，重点是在这些新兴研究领域的研究生和博士后的培训以及分析中使用的各种技术。这些努力将促进下一代数学家，这对国家科学基础设施至关重要。