Let R be a local ring and I an ideal of finite colength in R. We assume that I is integrally closed. In this talk, I will discuss an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if R has sufficiently nice singularities. I will explain the proofs for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. This is joint work with Ilya Smirnov. Speaker(s): Hai Long Dao (University of Kansas)