Multilinear Singular Integrals and Applications

Harmonic analysis studies mathematical objects through their decomposition into elementary waves. It is a central area of mathematics with applications in signal processing, medical imaging, communications technology, and quantum physics, while also providing essential tools used throughout many areas of pure mathematics. A core objective of harmonic analysis is the quantitative study of functions and signals and their behavior under various transformations, with frequency filtering being a basic example. Such questions are typically formulated in terms of inequalities involving quantities that measure size, regularity, or oscillatory behavior. This project aims to develop new inequalities for integral transformations that arise naturally in harmonic analysis and investigate their implications for problems in ergodic theory and related fields. Ergodic theory studies dynamical systems under repeated evolution and seeks to understand their asymptotic behavior. Central objects are ergodic averages, which encode statistical information obtained from repeated observation of a system over time. This work addresses convergence questions, with the aim of not only establishing convergence of such averages in various senses but also quantifying the rate of convergence. Related combinatorial problems on the structure of arithmetic and geometric configurations within large sets are also investigated. In addition to addressing foundational problems in harmonic analysis and adjacent fields, the project aims to incorporate computer-assisted verification of selected results, contributing to the growing field of formally verified mathematics. Such efforts increase the reliability of mathematical arguments and help build machine-verifiable libraries of results that can support future research. The project also involves organizing seminars and workshops that promote interaction among mathematicians and facilitate the exchange of ideas, as well as the training of students. The research program is organized around three interconnected themes. The first concerns establishing Lebesgue norm bounds for multilinear singular integrals within the broader framework of singular Brascamp-Lieb inequalities. These problems are studied in Euclidean spaces, with the aim of extending them to more general metric measure spaces. The second theme investigates applications of harmonic analysis techniques to multilinear ergodic averages along the orbits of multiple measure-preserving transformations. A central question, which has been studied in various settings, is whether such averages converge in norm or pointwise almost everywhere; pointwise convergence for general multiple commuting transformations remains a major open problem. In this direction, multilinear ergodic averages are studied using variation norm estimates to obtain quantitative convergence results in settings where qualitative convergence is available. The third theme concerns density theorems for point configurations in positive-density subsets of Euclidean space, using methods from multilinear singular and oscillatory integrals. The project also includes the formalization of selected arguments in Lean and research on harmonic analysis on the Hamming cube. 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. NSF Award ID: 2554859 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Polona Durcik | Institution: Chapman University, ORANGE, CA | Award Amount: $133,477 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554859 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554859.html