In this talk, I will discuss a recent result on collapsing constant scalar curvature (CSC) metrics. We prove that a sequence of CSC metrics that is collapsing with bounded curvature to a manifold can be perturbed to a sequence of nilpotent-invariant collapsing CSC metrics, under a natural assumption involving the eigenvalues of the drift Laplacian on the limiting manifold.  This proves a special case of a conjecture of Cheeger-Fukaya-Gromov. We also give some natural conditions on the limiting metric-measure space under which the eigenvalue assumption is automatically satisfied. The upshot is that we build the Cheeger-Fukaya-Gromov theory in the finite regularity setting and solve the perturbed Yamabe equation in the weak setting. This is a joint work with Jeff Viaclovsky.