Mathematical Sciences: Flag and Schubert Schemes - Classical, Generalized and Quantum

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This project is concerned with research related to Flag and Schubert schemes. The principal investigator will work on the following problems: (1) develop a standard monomial theory for Kac-Moody groups; (2) determine the multiplicity of a singular point on a Schubert variety; (3) develop a standard monomial theory for tangent cones at singular points on a Schubert variety; (4) determine the Kazhdan-Lusztig polynomials P(w,t) for w greater than or equal to t; (5) write the equations of the conormal bundle of a Schubert variety; (6) develop a theory of quantum Flag and Schubert schemes in the finite and affine cases. This project is concerned with the study of algebraic groups. It combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal field theory.