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（哥伦布校园）举行的关于谐波分析和分形几何学主题的会议。 分形是高度不规则的几何对象，在任意尺度上显示细节。 这种集合自然出现在数学中，并在科学和工程中的各种问题上都有应用。 谐波分析在20世纪后期出现，是研究分形集合的几何形状的工具，而分形几何形状促使了谐波分析中的许多有趣的问题。该事件汇集了研究人员，以发现现有方法之间的新协同作用，并制定新方法以在这些领域的交集中建立最新进展。会议的重点是使用谐波分析工具的分形集合的几何特性。该领域的一个经典问题是Falconer距离问题，它询问紧凑型集合的Hausdorff尺寸必须是确保其确定正面的Lebesgue度量集合集的大小。 更一般而言，人们可能会询问何时高维集确定更复杂的模式，从而导致有关有限点配置的许多有趣的问题。 可以通过卷积操作员的范围来研究这些问题，从而从谐波分析中带来熊工具。另一个经典的例子是Kakeya问题，该问题询问何时包含各个方向的线段的设置必须具有完整的Hausdorff尺寸。 Kakeya问题反过来与傅立叶限制理论紧密相关。在整个会议期间，反复出现的主题是傅立叶变换与结构概念之间的关系，例如Hausdorff Dimension.https：//u.osu.edu/hafs2023/this奖，这反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和广泛的影响来评估CRITERIA的评估。