Assess a graduate student's ability to design explicit families of compact Riemannian manifolds that are pairwise isospectral but not isometric while realizing a prescribed topology (e.g., a given diffeomorphism type or fundamental group). Problems should require producing explicit metrics/constructions (using tools such as Sunada/Gassmann triples, covers, torus or lens-space constructions, or transplantation), proving isospectrality, and giving rigorous non-isometry certificates (via length spectra, heat invariants, or geometric/topological obstructions).