Time-Frequency Analysis for Curved Multilinear Singular Operators and Applications to Related Fields
Ente: ANALYSIS PROGRAM
Scadenza: 2029-07-31
Importo max: 205.376 EUR
Paese: US
Descrizione
This project in classical harmonic analysis studies multilinear operators with curvature and their connections with additive combinatorics and ergodic theory. Multilinear operators describe interactions among several functions, while curvature reflects the nonlinear geometric structure present in the underlying singular integrals, maximal operators, and oscillatory averages. Such problems are central in modern Fourier analysis and provide model cases for understanding how oscillation, cancellation, and geometry interact. Progress on these problems has the potential to clarify several difficult questions in harmonic analysis and to provide tools for related problems involving pointwise convergence, nonlinear patterns, and Fourier-analytic methods in additive combinatorics and ergodic theory. The project also supports education and dissemination through graduate courses, seminars, mentoring of undergraduate and graduate students, mentoring of postdoctoral researchers, and publicly available research and teaching materials. Outreach activities at Auburn University include participation in programs such as Destination STEM, the Auburn University Summer Science Institute, and the Alabama Science and Engineering Fair, together with the development of interactive educational materials that make mathematical ideas accessible to students and the broader public.
The project consists of two closely related parts. The first part studies boundedness problems for multilinear singular and maximal operators whose defining geometry contains nonzero curvature. This includes higher-order multilinear Hilbert transforms, higher-order triangular Hilbert transforms, curved trilinear Hilbert transforms beyond previously established ranges, and bilinear Hilbert-Carleson type operators in curved settings. The goal is to develop methods that capture the cancellation created by curvature and that extend the current theory from bilinear or trilinear model cases to higher-order multilinear problems. The second part applies these ideas to related questions in harmonic analysis, additive combinatorics, and ergodic theory. These include multilinear spherical maximal operators, nonlinear Roth-type theorems for longer power-type patterns, and pointwise convergence of certain weighted continuous-time ergodic averages. Together, these directions aim to deepen the understanding of curved multilinear phenomena and to establish new links between time-frequency analysis, nonlinear pattern problems, pointwise convergence questions, and Fourier-analytic methods.
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.
Istituzione: Auburn University
Sede: AUBURN, AL
PI: Bingyang Hu
Settori: Mathematical & Physical Sciences
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