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

奇异积分是在偏微分方程研究中占有重要地位的数学对象，其应用范围从物理到工程再到量子计算。奇异积分的数学理论传统上是在光滑的几何环境中表述的。然而，随着对不规则或非光滑环境中物理现象的数学模型的更精细理解，对粗糙环境中奇异积分理解的需求最近有所增长。奇异积分在量子计算中的新兴应用进一步支持了对经典理论扩展的需求。值得注意的是，奇异积分与集合和测度几何之间的关系促进了对高维点集降维的新理解，即检测高维空间中的大集合点是否实际上位于光滑的低维流形上的机制。这种性质的结果对于数据科学应用非常重要，该项目有可能将奇异积分理论的工具包带到这个重要的应用领域。通过将纯谐波分析方法与组合学和概率论的工具相结合，并通过其与数据科学相关问题的显著接口，该项目还将为初级数学家（包括研究生）的培训提供机会。该项目考虑了在非光滑或粗糙设置下奇异积分研究中的各种问题，使用现有和新开发的工具。主要研究人员一直站在这种理论过去发展的前沿，目前的项目将使几何和分析的其他领域的新应用具体化。本课题考虑的问题包括：(a)具有矩阵权的有界奇异积分的尖锐表征，这在向量平稳随机过程的正则性理论中是重要的；(b)具有环的图（多树，Hamming立方等）上的副积奇异算子的加权有界性的表征；(c)大于1的余维的David-Semmes正则性问题。后一个主题将该项目与几何测量理论中的问题和降维研究联系起来，并伴随着对大数据集几何的影响。该奖项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知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.