Dalimil Mazac, of the Institut de Physique Theorique at CEA Paris-Saclay, presents this research seminar at the Institute for Advanced Study's Special Analysis and Mathematical Physics program. He reviews the conformal bootstrap, a method that has proven powerful for analyzing reflection-positive conformal models, and then proposes a way to extend it beyond that reflection-positive setting. His approach draws on the spectral theory of automorphic forms, treating the bootstrap problem as analogous to decomposing functions on arithmetic spaces into eigenfunctions. The talk assumes graduate-level familiarity with conformal field theory and representation theory, moving through the technical machinery needed to connect probabilistic conformal models to automorphic spectral methods. Delivered from Simonyi Hall at IAS, it is a specialist mathematical physics lecture aimed at researchers already working in conformal bootstrap or automorphic form theory rather than a general audience introduction.