Certain Algebraic Varieties Related to Finite and Affine Flag Varieties

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Flag varieties form an important class of varieties in AlgebraicGeometry. Schubert varieties form an important class of subvarieties insideflag varieties. The principal investigator developed a ``Standard MonomialTheory" (henceforth abbreviated SMT) in collaboration with Musili andSeshadri for flag varieties and their Schubert varieties. This theory hasled to very many important geometric & representation-theoreticconsequences. The principal investigator's recent research shows that oncethere is a good SMT for an algebraic variety, much information could beinferred about the variety (using SMT); for instance, SMT throws light onthe degenerations of the variety. The degenerations of a variety in turnfacilitate the understanding of the geometric aspects of the variety. Thistechnique has been used very recently in the area of Complexity Theory inComputer Science, esp., in the context of the ``P not equal toNP-conjecture". There is yet another interesting and important class ofalgebraic varieties related to Flag varieties, namely, the class of orbitclosures for the Adjoint action of a semi-simple algebraic group on thevariety of nilpotent elements in its Lie algebra. While there are very manyinteresting algebraic varieties - the determinantal varieties, Ladderdeterminantal varieties, quiver varieties - which get identified in anatural way with certain open subsets in Schubert varieties, theabove-mentioned orbit closures (and more generally orbit closures arisingfrom cyclic quivers) get identified in a natural way with certain opensubsets in affine Schubert varieties, i.e., Schubert varieties in thegeneralized flag variety associated to a Kac-Moody algebra. The researchproject considered in this proposal aims at developing a SMT for theabove-mentioned orbit closures via the fore-said relationship with Schubertvarieties ; it also aims at developing a SMT for other interesting classesof varieties - large Schubert varieties, Spherical varieties etc., as wellas determining in an explicit way the multiplicative structure of theequivariant Grothendieck ring and the equivariant cohomology ring of flagvarieties.Modern Algebraic Geometry (developed in the later half of the 20-th century)has proved itself (beyond any doubts) to be indispensable in variousdisciplines within Mathematics as well as in other areas outsideMathematics:Examples: Topology, Representation Theory, Combinatorics (withinMathematics).The modern Quantum Theory (especially Quantum & Conformal field theories) inPhysics.Robotics, Complexity Theory, Computer vision in Computer Science.This proposal is at the cross-roads of Commutative Algebra, AlgebraicGeometry, Combinatorics & Representation-theory. The varieties studied inthis proposal form an important class of varieties in Algebraic Geometry;for example, the theory of Schubert varieties (over finite fields) isclosely linked to Coding theory. The principal investigator believes thatthis proposal is bound to have significant impacts on the above-mentioneddisciplines.