In this talk I will define what is a Riemann-Hilbert problem, give an elementary example. Then I will present the solution of the Riemann-Hilbert that gives rise to solution of Painlevé-I equation. Next I will show the Schlesinger transformation of this solution, which will lead to a different solution.

 

In the end I will discuss how these two solutions are related to the universality phenomenon in nonlinear wave equations, and the Dubrovin universality conjecture.

 

This talk is intended for general mathematics audience. No specialized knowledge is assumed. I will walk you through concrete examples and elementary computations.