Random functions and stochastic processes on random graphs

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.