I will present conjectures and results concerning intersection theoretic invariants of parameter spaces of sheaves over low dimensional varieties, with emphasis on the Quot schemes of curves and surfaces.

A Schubert variety in the complete flag variety $GL_n/B$ is Levi-spherical if the action of a Borel subgroup in a Levi subgroup of a standard parabolic has an open dense orbit. I will present some recent work, joint with Alexander Yong and Yibo Gao, giving a classification of Levi-spherical Schubert varieties in terms of spherical elements of a symmetric group. I will also discuss a conjectural extension of the classification to other Lie types.