in the theory of von neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to sorin popa's breakthrough deformation/rigidity theory. popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, 'deformability' of an algebra $m$ allows one to precisely locate 'rigid' subalgebras known to exist only via a supposed isomorphism $m \cong n$. this talk will focus on joint work with rolando de santiago, ben hayes, and thomas sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. in particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. the case where this family has two members answers a question of jesse peterson asked at the american institute of mathematics (aim), but the result is most striking when the family is infinite.