Explore a comprehensive lecture on variational problems and spectral curves in the context of the hermitian matrix model with external source. Delve into the basic question, using the hermitian matrix model as a warm-up example. Gain an overview of key results, focusing on finite N spectral curves. Understand the relationship between spectral curves and variational problems, and receive insights into the proof methodology. This 44-minute talk, presented by Guilherme Lima Ferreira da Silva from the University of Michigan, is part of the School and Workshop on Random Matrix Theory and Point Processes at ICTP Mathematics.