Extensions and Applications of Fourier Decoupling Theory

North Carolina State University, Raleigh NC

Investigators

Abstract

How can we measure the cancellation that occurs when complex wave behavior is broken down into simpler, more manageable components? Is there a way to quantify the constructive and destructive interference produced as waves traveling in different directions interact? One way to answer this question is using Fourier decoupling, a mathematical theory that has recently led to spectacular advances on various longstanding conjectures across diverse areas of mathematics. The PI aims to broaden the scope of Fourier decoupling by extending it to new settings such as alternative ways of measuring cancellation and more general geometric surfaces currently not covered by the known theory. Another goal is to build a unifying framework between Fourier decoupling and the seemingly unrelated mathematical theory of efficient congruencing which will bring new ideas to the areas of Fourier analysis and number theory. Furthermore, the PI plans to apply techniques from Fourier decoupling to the analysis of Boolean functions in computer science, with the goal of deepening our theoretical understanding of areas such as algorithmic learning theory. Finally, the project will involve undergraduate and graduate students in research, the organization of international summer schools, and develop an early-career analysis community across the US advancing both scientific discovery and educational development. In more detail, the project focuses on advancing Fourier decoupling theory and its applications across harmonic analysis, number theory, and theoretical computer science. Key goals include: (1) interpreting the 2012 Parsell-Prendiville-Wooley argument on translation-dilation invariant systems in decoupling language; (2) developing new mixed norm decoupling inequalities relevant to dispersive partial differential equations and extending known Strichartz estimates; (3) adapting the high-low method to the moment curve setting to derive logarithmic power bounds for decoupling constants, giving potential number theoretic applications; (4) proving new refined decoupling estimates and applying them to problems like the Mizohata-Takeuchi conjecture; and (5) exploring connections between Euclidean harmonic analysis and Boolean function analysis, especially the Bohnenblust-Hille inequality and the Friedgut-Kalai-Naor Theorem, with implications for learning theory. Expected contributions include sharper bounds in decoupling theory, new decoupling theorems for highly general surfaces, improved maximal function bounds for the Schrödinger equation on the torus, and new analytic tools for use in algorithmic contexts. 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.