Topics in Harmonic Analysis and Probabilistic Analysis

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Many physical systems can be described using nonlinear equations, however it is often difficult to solve these equations explicitly and therefore challenging to locate and analyze the solutions. For example, in mathematical physics there are equations describing important phenomenon (such as propagation of light in nonlinear medium, propagation of waves in shallow water) where one would like to understand the long-time behavior of the solutions. Another example is from statistical physics, where one would like to understand the statistics of the solutions of random polynomial equations (often of very large degree), in particular it is of special interest to estimate and determine the typical locations of the solutions that are real. The purpose of this project is to further develop and employ tools in harmonic analysis to investigate various open questions related to these topics. The underlying theme of the project is the use of real variable techniques to study behavior of nonlinear equations (both deterministic and random) in the presence of a large parameter. The proposed research branches into two directions: (1) analysis of nonlinear Fourier transforms arising in inverse scattering theory and related nonlinear oscillatory integrals, and (2) analysis of the roots of random algebraic polynomials. For (1) the problems to be investigated are related to uniform boundedness of truncated nonlinear Fourier transforms and long-time asymptotics for nonlinear oscillatory integrals. For (2) the problems to be investigated are related to the estimation of several key statistics for the distribution of the real roots when the degree of the polynomial is large. The main tools developed and employed in the principal investigator's previous work (solo and joint with co-authors) continue to be refined and sharpened in this project: a novel outer-measure framework designed to treat delicate iterated Fourier integrals in the multilinear expansion of nonlinear Fourier transforms, a real variable approach to study long time asymptotics of oscillatory Riemann-Hilbert problems, and improved methods to study the distribution of the real roots of random polynomials with varying coefficients.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.