Conference on Harmonic Analysis and Fractal Sets

调和分析与分形集会议

• 批准号: 2247346
• 负责人: Alexander McDonald
• 金额: $ 2.26万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-04-01 至 2024-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247346&HistoricalAwards=false
• 关键词: Conference; Harmonic; Analysis; Fractal; Sets

The award supports a March 2023 conference at the Ohio State University (Columbus campus) on the topics of harmonic analysis and fractal geometry. Fractals are geometric objects which are highly irregular, exhibiting detail at arbitrarily small scales. Such sets arise naturally in mathematics, and have applications to a diverse array of problems in science and engineering. Harmonic analysis emerged in the late 20th century as a tool for the study of the geometry of fractal sets, and in return fractal geometry has motivated many interesting questions in harmonic analysis. This event brings together researchers to find new synergies between existing methods and to formulate new approaches to build on recent progress at the intersection of these fields.The focus of the conference is on geometric properties of fractal sets using tools from harmonic analysis. A classical problem in this area is the Falconer distance problem, which asks how large the Hausdorff dimension of a compact set must be to ensure that it determines a positive Lebesgue measure set of pairwise distances. More generally, one may inquire when high dimensional sets determine more complicated patterns, leading to a host of fascinating questions about finite point configurations. These problems can be studied via bounds for convolution operators, bringing to bear tools from harmonic analysis. Another classical example is the Kakeya problem, which asks when sets containing a line segment in every direction must have full Hausdorff dimension. The Kakeya problem, in turn, is closely tied to Fourier restriction theory. A recurring theme throughout the conference is the relationship between the Fourier transform and notions of structure such as Hausdorff dimension.https://u.osu.edu/hafs2023/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.

该奖项支持2023年3月在俄亥俄州立大学(哥伦布校区)举行的关于调和分析和分形几何主题的会议。分形图是高度不规则的几何对象，在任意小的尺度上显示细节。这样的集合自然出现在数学中，并应用于科学和工程中的各种问题。调和分析作为一种研究分形集几何的工具出现在20世纪后期，作为回报，分形几何激发了调和分析中许多有趣的问题。这次会议聚集了研究人员，以发现现有方法之间的新协同效应，并制定新的方法，以在这些领域的交集的最新进展的基础上再接再厉。会议的重点是使用调和分析工具研究分形集的几何性质。这一领域的一个经典问题是Falconer距离问题，它询问紧集的Hausdorff维度必须有多大才能确保它确定两两距离的正Lebesgue度量集。更广泛地说，人们可能会问，何时高维集合决定了更复杂的模式，导致了一系列关于有限点配置的有趣问题。这些问题可以通过卷积算子的界来研究，利用调和分析的工具。另一个经典的例子是Kakeya问题，它询问在每个方向上包含一条直线段的集合何时必须具有全Hausdorff维。反过来，Kakeya问题又与傅里叶限制理论密切相关。整个会议中反复出现的主题是傅里叶变换和结构概念之间的关系，例如豪斯多夫维度。https://u.osu.edu/hafs2023/This奖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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。