Modern Schubert Calculus studies various intersection rings associated to flag manifolds. All these rings have several common features: they all have a distinguished Schubert basis; the Schubert structure constants count certain intersection points; the Schubert classes can be defined by equivariant localization. A question with roots in representation theory and microlocal analysis is whether there are good analogues of Schubert classes to study intersection rings of the cotangent bundle of a flag manifold. One answer, coming from singularity theory, leads to the study of the Chern-Schwartz-MacPherson classes of Schubert cells. For flag manifolds, these classes are Schubert positive, and they are equivalent to the stable envelopes on the cotangent bundle, defined recently by Maulik and Okounkov. I will explain these ideas, and draw parallels with the Schubert Calculus situation. For instance, instead of counting points in three Schubert cycles, in the cotangent situation one takes the Euler characteristic of the intersection of three Schubert cells. Based on joint work with Paolo Aluffi, Jorg Schurmann and Changjian Su.