Topics in Harmonic Analysis: Time-frequency Analysis and connections with Additive Combinatorics and Partial Differential Equations

Purdue University, West Lafayette IN

Investigators

Abstract

This project lies mainly within the area of harmonic analysis, with a special focus on revealing the deep connections between time-frequency analysis and fields such as additive combinatorics, incidence geometry, and partial differential equations (PDE). The project investigates three major themes: 1) the problem of the pointwise convergence of Fourier series (near the integrability threshold), 2) the role of curvature in several model problems connected with major open problems including Zygmund's differentiation conjecture and the boundedness of the trilinear Hilbert transform, and 3) applications in PDE and fluid mechanics. The first theme of pointwise convergence lies at the very foundation of Fourier analysis and has a history of more than 200 years. Along the way, it has generated new areas within mathematical analysis (e.g. time-frequency analysis) and served as a motivation and inspiration for the development of wavelet theory -- a field that nowadays has numerous real-world applications in engineering (image processing, data recovery), medicine (MRI), and other fields. The second theme aims to understand toy curved models designed to shed new light on longstanding open problems with applications to ergodic theory and number theory. Finally, the third theme deals with questions such as the global behavior of the maximal Schrodinger operator and the study of singularity formation in two dimensions for water waves -- the latter having direct implications for our understanding of physical reality. This is a diverse project involving relevant problems in harmonic analysis, with applications in several related fields. The PI has obtained relevant progress on all three of the themes discussed above. Indeed, for the first theme, the PI completely solved the lacunary model of the problem, in particular verifying a conjecture posed by Konyagin at the 2006 International Congress of Mathematicians. Moreover, in the course of his work on the lacunary model, the PI established deep and surprising connections between time-frequency analysis and additive combinatorics, which he is now using to develop a new methodology for approaching the longstanding full problem. For the second theme -- meant to develop a deeper understanding of the role of curvature in harmonic analysis, by studying the transition from nonflat to flat models of some difficult well-known open problems -- the PI developed a program which recently unified three directions: the Hilbert transform, the bilinear Hilbert transform, and the Carleson operator along non-flat curves (together with their maximal variants). The third theme has strong connections with dispersive PDE and fluid mechanics. The fluid mechanics component is part of a joint project and aims to understand the asymptotic geometry of the interface between two fluids in a 2D setting as the interface approaches a "splash" scenario from the water-wave case. This project builds on the previous joint work of the PI together with C. Fefferman and A. Ionescu on the lack of splash singularity formation in the 2D case of locally smooth two-fluid interfaces under irrotational assumptions. 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.