A deformation of a local ring R is another local ring S such that R = S/f for some nonzero divisor f, and the deformation is called embedded if f lies in the square of the maximal ideal of S. Detecting the presence of embedded deformations is a subtle problem. Avramov observed that embedded deformations give rise to nontrivial central elements in the “homotopy Lie algebra” associated to R, andin 1989 he asked whether all central elements arise in this way. In this talk I will explain these ideas and present a counterexample to Avramov's question, which had remained open. I will also discuss connections with quasi-complete intersection homomorphisms and an old problem of Rodicio concerning their structure. This is joint work with Eloísa Grifo, Josh Pollitz, and Mark Walker