Compact hyperbolic surfaces of fixed topology are the subject, and Princeton mathematician Anshul Adve uses their Laplace eigenvalues and the structure constants of eigenform multiplication to build a converse theorem. The talk, given at the Joint IAS/Princeton Analysis Seminar, starts from the observation that these spectral numbers obey algebraic constraints resembling the conformal bootstrap equations used in physics. Adve's main result shows the relationship runs both ways: any collection of numbers satisfying the constraints must in fact arise from some hyperbolic surface, not just resemble one. He closes with applications to upper bounds on spectral gaps and subconvex bounds for L-functions, noting explicitly that no background in physics or L-functions is assumed. The blackboard talk moves through the constraint equations, the construction behind the converse theorem, and the spectral geometry connecting them, aimed at an audience with analysis and geometry background rather than general viewers. Fifty-eight minutes, single speaker, seminar room setting.