Asymptotic analysis of repulsive point processes and integrable equations

The proposed project will apply and develop Riemann-Hilbert (RH) methods for solving asymptotic questions in random matrix theory, tiling models and nonlinear integrable partial differential equations.

The proposal has three main objectives: 1) obtain new asymptotic formulas of large structured determinants arising in random matrix theory, such as Toeplitz, Hankel, Muttalib-Borodin and Ginibre determinants, 2) develop a new approach to the asymptotic analysis of non-Hermitian matrix-valued orthogonal polynomials and use it to solve some universality conjectures in tiling models, 3) solve the long-standing problem of obtaining the long-time asymptotics of the solution of the bad Boussinesq equation.To solve these problems, one of the tools we will use is the Deift-Zhou steepest descent method.

This method has grown considerably over the last 25-years, and is now considered one of the most powerful tools of asymptotic analysis.

By identifying new areas where RH methods can be brought to bear and by solving new problems using this approach, this project will contribute to the development of the method itself.

Since the range of applicability of RH methods is very broad, these new techniques are likely to have an impact on a wide spectrum of scientific questions.