Title: A numerical Riemann-Hilbert approach to the computation of transform pairs

Abstract: This talk introduces a unified framework that integrates spectral theory, Riemann–Hilbert problems, and inverse scattering theory to construct and numerically evaluate transform pairs for time-evolution PDEs with variable coefficients. The method combines explicit analytical formulae with numerical techniques for ordinary differential equations and Riemann–Hilbert problems, producing a hybrid analytical–numerical strategy for studying such equations. The ideas will first be illustrated through the classical Fourier transform. Preliminary results are then presented for transforms associated with the Dirac equation, highlighting the potential of the approach to achieve accurate and stable computations, even in the presence of discontinuous coefficients.