This research will concentrate on problems connected with a semi-simple, simply-connected algebraic group G over a field k, the Borel subgroup B of G and a Schubert variety in the flag variety G/B. The P.I. has developed a standard monomial theory as a generalization of the classical Hodge-Young standard monomial theory. The following specific problems will be considered. Determine the multiplicity of a singular point on a Schubert variety. Write down the equations of the conormal bundle of a Schubert variety. Study the variety XY = YX = 0 where X and Y are matrices. Develope a standard monomial theory for a Kac group associated to an indecomposable, symmetrizable, generalized Cartan matrix of affine type. This research is in algebraic geometry, the study of the geometric objects (varieties) arising as solutions of systems of polynomial equations. The P.I. considers very concrete varieties and asks for very explicit, combinatorial answers to them. The types of varieties she considers are very important and the type of answers she seeks are widely applicable. Very important results will come out of this research.
