Varieties with group actions

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The main theme of this proposal is the study of algebraic varieties with algebraic group actions, typical examples being flag varieties. Homogeneous spaces form basic fundamental objects in Geometry and other related fields. The flag varieties constitute an important class of homogeneous spaces; Schubert subvarieties in flag varieties provide a powerful inductive machinery for the study of flag varieties. Problems of this proposal are related to some interesting and important algebraic varieties - Schubert varieties, affine Grassmannians (and affine Schubert varieties), quiver varieties, nilpotent orbit closures, toric varieties. Lakshmibai, Musili and Seshadri developed a "Standard Monomial Theory" (henceforth abbreviated SMT) for flag varieties and their Schubert subvarieties. This theory has led to very many important geometric & representation-theoretic consequences. This proposal deals first with developing a SMT for the varieties mentioned above, next proposes to use SMT to study such geometric problems as determination of singular locus, determination of the multiplicity at a singular point etc. Lakshmibai'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".Modern Algebraic Geometry (developed in the latter 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, such as: Topology, Representation Theory, Combinatorics (within Mathematics), the modern Quantum theory (especially Quantum & Conformal field theories) in Physics; Robotics, Complexity theory 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.