Explore the foundations of scattering data convergence and non-linear Fourier transform in this 55-minute mini-course lecture delivered by Alexei Poltoratski from the University of Wisconsin–Madison. Dive into topics such as canonical Hamiltonian systems, branch space definitions, and Hilbert spaces of entire functions. Examine spectral problems for canonical systems and delve into representation kernel Hilbert spaces. Gain insights into the historical context and key concepts of this mathematical field as part of the Focus Program on Analytic Function Spaces and their Applications at the Fields Institute.