Assess students' ability to analyze spectral properties of non-self-adjoint differential operators: define and compute pseudospectra, relate resolvent norm estimates to spectral instability, derive bounds connecting pseudospectra to growth/decay of evolution semigroups (using Gearhart–Prüss, Hille–Yosida, and pseudospectral abscissa), evaluate model examples (convection–diffusion, complex potentials), discuss numerical range and spectral pollution, and apply semiclassical or Carleman techniques for resolvent estimates. Problems require computing pseudospectra/resolvent behavior, proving semigroup stability or instability, and interpreting long-time dynamics.