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Topics in Harmonic Analysis and Probabilistic Analysis

谐波分析和概率分析主题

基本信息

批准号：
1800855
负责人：
Yen Do
金额：
$ 15.35万
依托单位：
University of Virginia Main Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2022-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800855&HistoricalAwards=false
关键词：
Topics Harmonic Analysis Probabilistic

项目摘要

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.

许多物理系统都可以用非线性方程来描述，但是通常很难显式求解这些方程，因此定位和分析解具有挑战性。例如，在数学物理学中，有一些方程描述了一些重要的现象（如光在非线性介质中的传播，波在浅水中的传播），人们希望了解这些方程解的长期行为。另一个例子来自统计物理学，其中人们希望了解随机多项式方程（通常是非常大的次数）的解的统计，特别是估计和确定真实的解的典型位置是特别感兴趣的。这个项目的目的是进一步开发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真实的变量技术来研究存在大参数的非线性方程（确定性和随机性）的行为。本文的研究主要分为两个方向：（1）分析逆散射理论中的非线性傅里叶变换和相关的非线性振荡积分;（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Fourier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次数较大时真实的根分布的几个关键统计量的估计。主要研究者以前工作中开发和使用的主要工具（单独和与共同作者联合）继续在本项目中得到完善和加强：一种新的外测度框架，用于处理非线性傅里叶变换的多线性展开中的精细迭代傅里叶积分，一种真实的变量方法，用于研究振荡Riemann-Hilbert问题的长时间渐近性，和改进的方法来研究具有不同系数的随机多项式的真实的根的分布。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。