Saverio Spagnolie, Fall 2020:

This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. The course introduces advanced mathematical methods with a particular focus on asymptotic analysis of algebraic, differential, and integral equations, including the methods of stationary phase, steepest descent, multiple scales, matched asymptotics, and WKB approximations. Other topics include divergent series, Fourier transform, complex integration, dimensional analysis, phase-plane analysis, Floquet theory, Green's functions, self-adjoint operators, Sturm-Liouville problems, and calculus of variations.