Collaborative Research: Non-homogeneous Harmonic Analysis, Spectral Theory, and Weighted Norm Estimates

合作研究：非齐次谐波分析、谱理论和加权范数估计

• 批准号: 2154402
• 负责人: Alexander Volberg
• 金额: $ 29.83万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154402&HistoricalAwards=false
• 关键词: Collaborative; Research; Non; homogeneous; Harmonic

Singular integrals are mathematical objects that feature heavily in the study of partial differential equations, with applications ranging from physics to engineering to quantum computing. The mathematical theory of singular integrals has traditionally been formulated in smooth geometric settings. However, demand for an understanding of singular integrals in rougher settings has grown recently with a more refined understanding of mathematical models for physical phenomena in irregular or non-smooth environments. Emerging applications of singular integrals in quantum computing further buttress the need for such extensions of the classical theory. Notably, the relationship between singular integrals and the geometry of sets and measures facilitates a new understanding of dimension reduction for high-dimensional point sets, that is, mechanisms to detect whether large collections of points in a high-dimensional space in fact lie on a smooth lower-dimensional manifold. Results of this nature are important for data science applications, and the project has the potential to bring the toolkit of singular integral theory to bear on this important application domain. By coupling pure harmonic analysis methods with tools from combinatorics and probability, and through its noticeable interface with questions of relevance in data science, the project will also provide opportunities for the training of junior mathematicians, including graduate students.This project considers a variety of questions in the study of singular integrals in non-smooth or rough settings, using both existing and newly developed tools. The principal investigators have been at the forefront of the past development of such a theory, and the current project will crystallize new applications to other areas of geometry and analysis. Questions under consideration in this project include: (a) a sharp characterization of bounded singular integrals with matrix weight, which is important in the regularity theory of vector stationary stochastic processes, (b) a characterization of weighted boundedness for para-product singular operators on graphs with cycles (multi-trees, Hamming cubes, etc.), and (c) the David-Semmes regularity problem in codimensions larger than one. The latter topic ties the project to questions in geometric measure theory and to the study of dimension reduction, with concomitant implications for the geometry of large data sets.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.

奇异积分是数学对象，在偏微分方程的研究中具有很大的特征，其应用从物理学到工程到量子计算。传统上，奇异积分的数学理论是在平滑的几何环境中提出的。然而，最近对更粗糙的环境中奇异积分的理解的需求已经增长，对在不规则或非平滑环境中物理现象的数学模型有了更精致的了解。奇异积分在量子计算中的新兴应用进一步支撑了对经典理论的这种扩展的需求。值得注意的是，单数积分与集合和测量的几何形状之间的关系有助于对高维点集的维数的新理解，即检测高维空间中大量点的机制实际上是否在于平滑的低维级别。这种性质的结果对于数据科学应用很重要，并且该项目有可能使单数积分理论的工具包在这个重要的应用领域上。通过将纯谐波分析方法与组合学和概率的工具结合在一起，并通过其明显的界面与数据科学中的相关性问题结合在一起，该项目还将为培训初级数学家（包括研究生）提供机会。该项目认为，使用现有和新开发的工具在非智能或粗糙的环境中，使用现有和开发的工具来研究各种问题。主要研究人员一直处于过去发展的最前沿，当前项目将使新的应用结晶到几何和分析的其他领域。 Questions under consideration in this project include: (a) a sharp characterization of bounded singular integrals with matrix weight, which is important in the regularity theory of vector stationary stochastic processes, (b) a characterization of weighted boundedness for para-product singular operators on graphs with cycles (multi-trees, Hamming cubes, etc.), and (c) the David-Semmes regularity problem in codimensions larger than one.后一个主题将项目与几何措施理论的问题联系起来，并与降低维度的研究有关，对大型数据集的几何形状产生了影响。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准通过评估来通过评估来获得支持的。

项目成果

Leibniz International Proceedings in Informatics (LIPIcs):15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

莱布尼茨国际信息学会议录 (LIPIcs)：第 15 届理论计算机科学创新会议 (ITCS 2024)

• DOI: 10.4230/lipics.itcs.2024.16
• 发表时间: 2024
• 期刊: Leibniz international proceedings in informatics
• 作者: Blackwell, Keller;Wootters, Mary
• 通讯作者: Wootters, Mary

Tail spaces estimates on Hamming cube and Bernstein–Markov inequality

汉明立方和伯恩斯坦马尔可夫不等式的尾空间估计

• DOI: 10.1016/j.jmaa.2023.127597
• 发表时间: 2024
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Volberg, Alexander

The Buffon's needle problem for random planar disk-like Cantor sets

随机平面盘状康托集的布丰针问题

• DOI: 10.1016/j.jmaa.2023.127622
• 发表时间: 2024
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Vardakis, Dimitris;Volberg, Alexander
• 通讯作者: Volberg, Alexander