Explore the computational theory of Riemann-Hilbert problems in this advanced mathematics lecture by Thomas Trogdon. Delve into topics such as computing Cauchy integrals, controlled bases, generalized contours, and singular integral equations. Examine Sobolev spaces, zero-sum spaces, and the regularity of jump matrices. Learn about numerical solutions for Riemann-Hilbert problems and their applications to nonlinear equations like the defocusing nonlinear Schrödinger equation and the KdV equation. Gain insights into steepest descent methods, code implementation, and various deformations. This in-depth lecture is part of a program on integrable systems in mathematics, condensed matter, and statistical physics organized by the International Centre for Theoretical Sciences.