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

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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.