Analysis and applications of orthogonal polynomials with zeros in the complex plane.

The topic of the research project is a classical one: the determination of zeros of polynomials in the complex plane. Despite being an old problem, several important research questions remain unanswered in this setting. Quite surprisingly, the study of the polynomials in this project is a shared problem in two different areas of current and highly active international research: the theory of random matrices and the theory of highly oscillatory integrals. Both areas in turn have applications in diverse disciplines of science. Random matrices provide a statistical description of many physical systems and highly oscillatory integrals appear, e.g., in the numerical simulation of wave phenomena, such as those found in electromagnetics, quantum physics or acoustics. The methodology of the project is based on a modern tool of analysis, namely the nonlinear steepest descent method for the asymptotic analysis of Riemann-Hilbert problems. Promising results have been obtained by the promotor and co-promotor using this approach for an initial special case. It is expected that these results can be significantly extended and generalized, with relevance simultaneously in both application areas. In particular, the proposed research contributes to the understanding and analysis of random matrices with external source and coupled random matrices. In addition, it contributes to the analysis and construction of numerical methods for the evaluation of highly oscillatory integrals based on Gaussian quadrature.

Keywords:Orthogonal polynomials, Asymptotic analysis, Numerical integration, Riemann-Hilbert problem, Steepest descent method, Oscillatory integrals, Random matrices

Disciplines:Applied mathematics in specific fields, Computer architecture and networks, Distributed computing, Information sciences, Information systems, Programming languages, Scientific computing, Theoretical computer science, Visual computing, Other information and computing sciences, Analysis