Shrinking Ricci solitons are generalizations of positive Einstein manifolds which arise naturally in the analysis of singularities of the Ricci flow. At present, all known complete noncompact examples either split locally as products or possess conical structures at infinity. I will describe recent joint work with Lu Wang in which we prove that such conical structures admit little flexibility: if two shrinking solitons are asymptotic along some end of each to the same regular cone, then the solitons must actually be isometric on some neighborhoods of infinity of these ends. As an application, we prove that the only complete connected shrinking soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton.