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.

摘要对于0456538 Lacey、0456306 Iosevich和0456490 Magya，拟议的活动将集中于调和分析中当前感兴趣的一组主题：Kakeya猜想、傅立叶限制问题和Carleman估计、波动方程的平滑猜想以及Zygmund关于沿Lipschitz线族的微分的猜想。同样核心的是正在识别和研究的新问题，包括经典算子的离散模拟、Fuglede猜想的各个方面、具有退化正则关系的傅立叶积分算子以及复分析中的新类奇异积分。这些学科过去的进展受到了诸如组合学、加法组合学和数论等学科的影响，仅举几个突出的领域。反过来，这些问题也在为这些领域做出贡献。这项提案所允许的各项努力的协调应能加快在这一广泛议题上的进展。拟议的活动涉及中心问题，这些问题将导致新的分析技术模式，这些新模式涉及更高维度的波的行为问题，以及离散物体(即数字物体和连续物体)之间微妙区别的不同方面。此外，这些项目将借鉴一系列不同数学领域的方法和技术。这门学科的分析方法的广度和复杂程度为这些领域之间的相互关系提供了新的线索。它也可能成为继续研究的障碍。这个项目的一个重要重点是培训研究生和博士后在这些新兴的研究领域，以及在他们的分析中使用的各种技术。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。