Distribution of Roots of Random Functions

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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.