Random functions and stochastic processes on random graphs

随机系数多项式的根分析是一个不断发展的领域，在数学的子领域中有广泛的应用，包括近似理论、数学物理和常微分方程。这个项目将研究根数分布的各种特征。这个项目的另一个重点是随机网络上的接触过程，旨在了解传染的动态，因为它通过复杂的相互关联的系统传播。这项工作将提供对现实世界流行病和信息传播的见解，从而制定更有效的战略来控制和预防感染或思想的传播。本研究项目旨在加深我们对这些无处不在的随机结构的理解，并探索它们在现实问题中的应用。作为该项目的一部分，研究生和本科生将接受指导，研究成果将通过出版物和研究讲座传播给广泛的受众。更具体地说，本研究项目将研究实根数的方差和高矩的普遍性，以及该数的渐近分布。为了实现这一目标，将使用各种通用性技术来开发新的工具和连接。在接触过程方面，本项目特别关注易感者-受感染者-易感者（SIS）模型，并将探索生存时间的相变。将采用新颖的思路和方法来严格分析SIS接触过程。通过这些项目，不同领域之间的新联系有望出现，从而带来新的见解和应用。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。

The analysis of roots of polynomials with random coefficients is a growing area with applications in subfields across mathematics, including approximation theory, mathematical physics, and ordinary differential equations. This project will investigate various features of the distribution of the number of roots. Another focus of this project is the contact process on random networks, aiming to understand the dynamics of contagion as it spreads through complex interconnected systems. This work will provide insight into real-world epidemics and information dissemination, leading to the development of more effective strategies for controlling and preventing the spread of infections or ideas. This research project seeks to deepen our understanding of these ubiquitous random structures and to explore their applications in real-world problems. Graduate and undergraduate students will be mentored as part of this project, and the research findings will be disseminated through publications and research talks, reaching a wide audience.More specifically, this research project will investigate the universality of variances and higher moments of the number of real roots, along with the asymptotic distribution of this number. To achieve this, various universality techniques will be used to develop new tools and connections. Regarding the contact process, the project specifically focuses on the susceptible-infected-susceptible (SIS) model and will explore the phase transition of the survival time. Novel ideas and methods will be pursued to rigorously analyze the SIS contact process. Through these projects, new connections between different areas are anticipated to emerge, leading to fresh insights and applications.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.