Flag varieties form an important class of varieties in Algebraic Geometry. Schubert varieties form an important class of subvarieties inside flag varieties. The principal investigator developed a ``Standard Monomial Theory" (henceforth abbreviated SMT) in collaboration with Musili and Seshadri for flag varieties and their Schubert varieties. This theory has led to very many important geometric & representation-theoretic consequences. The principal investigator's recent research shows that once there is a good SMT for an algebraic variety, much information could be inferred about the variety (using SMT); for instance, SMT throws light on the degenerations of the variety. The degenerations of a variety in turn facilitate the understanding of the geometric aspects of the variety. This technique has been used very recently in the area of Complexity Theory in Computer Science, esp., in the context of the ``P not equal to NP-conjecture". There is yet another interesting and important class of algebraic varieties related to Flag varieties, namely, the class of orbit closures for the Adjoint action of a semi-simple algebraic group on the variety of nilpotent elements in its Lie algebra. While there are very many interesting algebraic varieties - the determinantal varieties, Ladder determinantal varieties, quiver varieties - which get identified in a natural way with certain open subsets in Schubert varieties, the above-mentioned orbit closures (and more generally orbit closures arising from cyclic quivers) get identified in a natural way with certain open subsets in affine Schubert varieties, i.e., Schubert varieties in the generalized flag variety associated to a Kac-Moody algebra. The research project considered in this proposal aims at developing a SMT for the above-mentioned orbit closures via the fore-said relationship with Schubert varieties ; it also aims at developing a SMT for other interesting classes of varieties - large Schubert varieties, Spherical varieties etc., as well as determining in an explicit way the multiplicative structure of the equivariant Grothendieck ring and the equivariant cohomology ring of flag varieties.
Modern Algebraic Geometry (developed in the later half of the 20-th century) has proved itself (beyond any doubts) to be indispensable in various disciplines within Mathematics as well as in other areas outside Mathematics: Examples: Topology, Representation Theory, Combinatorics (within Mathematics). The modern Quantum Theory (especially Quantum & Conformal field theories) in Physics. Robotics, Complexity Theory, Computer vision in Computer Science. This proposal is at the cross-roads of Commutative Algebra, Algebraic Geometry, Combinatorics & Representation-theory. The varieties studied in this proposal form an important class of varieties in Algebraic Geometry; for example, the theory of Schubert varieties (over finite fields) is closely linked to Coding theory. The principal investigator believes that this proposal is bound to have significant impacts on the above-mentioned disciplines.
