Formulas for the integrals of top Segre classes of tautological bundles over the Hilbert scheme of points of surfaces were conjectured in closed form by Lehn in 1999. I will explain a proof of this conjecture for K-trivial surfaces. Key to the argument are a set of recursions obtained by equivariant localization of the virtual class of a suitable Quot scheme. The recursions determining the Segre integrals fit into a much wider theory aimed at the study of tautological classes over the moduli space of K3 surfaces. This is based on joint work with Alina Marian and Rahul Pandharipande. Speaker(s): Dragos Oprea (UCSD)