Distribution of Roots of Random Functions

随机多项式自然而然地出现在物理和数学的各个领域，如量子混沌系统和逼近理论中。随机多项式的研究在计算机科学和工程中有着广泛的应用。此外，研究高次多项式的根是一个重要的数学领域，在理论科学和应用科学中都很有用。这个研究项目的目的是研究关于随机多项式的根的分布的基本问题，更广泛地说，随机函数的根的分布。这个项目包含三个研究项目，解决随机函数领域的问题。在第一个程序中，研究者计划建立随机正交多项式的局部普适性，并推导出一般分布的实根的平均个数。第二个程序旨在研究各种经典随机函数模型的实根个数的方差和中心极限定理。第三个程序的目标是进一步了解系数的增长和实根数的增长之间的联系。为了解决这些问题，研究人员将发展局部普适性方法，并在分析和概率方面建立不同的工具。该项目还将解决有关马尔可夫链的混合时间和接触过程的相变的几个问题，这是疾病在社区中传播的重要模型。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.