Emanuel Milman of the Technion Israel Institute of Technology presents this joint IAS/Princeton University Analysis Seminar talk on the mathematics of soap-bubble clusters. He builds a spectral analysis of the Jacobi operator on the interfaces where bubble walls meet, which by Plateau's laws always happens in threes at 120-degree angles, producing a self-adjoint operator whose spectrum determines whether a cluster is stable under volume-preserving perturbations. Milman frames stability as a local, infinitesimal version of the isoperimetric problem, equivalent to a Poincare-type inequality on the cluster. He reports a proof that standard k-bubble clusters in Euclidean, spherical, and hyperbolic spaces are stable for all dimensions n at least 3, extending the result to Mobius-flat partitions and to flat-interface partitions under Gaussian weighting. The argument rests on a new conjugated Brascamp-Lieb inequality for partitions with conformally flat umbilical boundary. The talk is dense graduate-level geometric analysis delivered to a specialist seminar audience, with blackboard-style derivation throughout.