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.

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.