Distribution of Roots of Random Functions

随机函数的根的分布

• 批准号: 2211929
• 负责人: Oanh Nguyen
• 金额: $ 17.45万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2024-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2211929&HistoricalAwards=false
• 关键词: Distribution; Roots; Random; Functions

Random polynomials occur naturally in various areas of physics and mathematics, such as in quantum chaotic systems and approximation theory. The study of random polynomials has applications in computer science and engineering. In addition, studying roots of high-degree polynomials is an important area of mathematics that is useful in both pure and applied sciences. The goal of this research project is to study fundamental questions concerning the distribution of roots of random polynomials and, more generally, random functions. This project contains three research programs that address questions in the field of random functions. In the first program, the investigator plans to establish local universality for random orthogonal polynomials and derive the mean number of real roots for general distributions. The second program aims to study the variance and the Central Limit Theorem for the number of real roots of various classical models of random functions. The goal of the third program is to further the understanding of the connection between the growth of the coefficients and the growth of the number of real roots. To approach these questions, the investigator will develop the local universality method and build on different tools in analysis and probability. The project will also address several questions regarding the mixing time of Markov Chains and the phase transition of the contact process, an important model for the spread of diseases in communities.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.

随机多项式自然存在于物理和数学的各个领域，例如量子混沌系统和近似理论。随机多项式的研究在计算机科学和工程中有应用。此外，研究高度多项式的根是数学的重要领域，在纯科学和应用科学方面都有用。该研究项目的目的是研究有关随机多项式根和更一般的随机功能的基本问题。 该项目包含三个研究计划，这些计划解决了随机功能领域的问题。在第一个程序中，研究人员计划为随机正交多项式建立本地普遍性，并得出一般分布的实际根平均数量。第二个程序旨在研究各种经典函数模型的真实根数的差异和中心限制定理。第三个计划的目的是进一步了解系数增长与实际根数的增长之间的联系。为了解决这些问题，研究者将开发本地普遍性方法，并以分析和概率的不同工具为基础。该项目还将解决有关马尔可夫链的混合时间以及接触过程的相位过渡的几个问题，这是一个在社区中疾病传播的重要模型。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子优点和更广泛的影响标准通过评估来进行评估。