Topics in Harmonic Analysis and Probabilistic Analysis

许多物理系统可以用非线性方程来描述，但是通常很难明确地求解这些方程，因此很难找到和分析解。例如，在数学物理中，有一些描述重要现象的方程（如光在非线性介质中的传播，波在浅水中的传播），人们希望了解解的长期行为。另一个例子来自统计物理学，人们想要了解随机多项式方程（通常是非常大的程度）解的统计信息，特别是对估计和确定实数解的典型位置特别感兴趣。该项目的目的是进一步发展和使用谐波分析工具来调查与这些主题相关的各种开放问题。该项目的基本主题是使用实变量技术来研究存在大参数的非线性方程（确定性和随机）的行为。提出的研究方向分为两个方向：(1)分析反散射理论和相关非线性振荡积分中产生的非线性傅立叶变换；(2)分析随机代数多项式的根。对于(1)所要研究的问题涉及截断非线性傅立叶变换的一致有界性和非线性振荡积分的长时间渐近性。对于(2)所要研究的问题涉及多项式的次较大时实根分布的几个关键统计量的估计。在这个项目中，主要研究者在以前的工作中开发和使用的主要工具（单独或与共同作者）继续得到完善和完善：在非线性傅里叶变换的多重线性展开中，提出了处理精细迭代傅里叶积分的新外测度框架，研究振荡Riemann-Hilbert问题长时间渐近性的实变量方法，改进了研究变系数随机多项式实根分布的方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.