Some Algebraic Varieties Related to Flag Varieties

9971295本文讨论了与齐次空间、Flag簇及其Schubert簇有关的一些重要的代数簇的问题。在Musili和Seshadri的合作下，P.I.通过发展--现在众所周知的--标准单项理论，在国旗品种几何领域取得了重大突破。这一理论是20世纪50年代S发展起来的经典Hodge-Young理论的推广。这一理论已经导致了大量的几何、表示论和组合结果。利用她在标准单体理论方面的工作，P.I.解决了许多关于Schubert簇的重要问题-几何和表示理论。最近，P.I.成功地将她的结果应用于某些代数簇的研究--某些环簇、阶梯行列式簇和箭图簇。齐次空间是几何中的基本对象。标志簇构成了一类重要的齐性空间。由于其丰富的几何学和组合学，FLAG簇在代数几何、代数群、表示论和组合学领域形成了有趣和重要的研究对象。旗帜品种中的舒伯特品种为旗帜品种的研究提供了有力的工具。

9971295This proposal deals with some problems on some important algebraic varieties related to homogeneous spaces, Flag varieties and their Schubert varieties. In collaboration with Musili and Seshadri, the P.I. made a major break-through in the area of the Geometry of the flag varieties by developing - what is now well-known as - the STANDARD MONOMIAL THEORY. This theory is a generalization of the classical HODGE-YOUNG THEORY developed in the 1950's. This theory has led to an astounding number of geometric, representation-theoretic and combinatorial consequences. Using her work on Standard Monomial Theory, the P.I. has solved many important problems on Schubert varieties - geometric and representation-theoretic. Recently, the P.I. has successfully applied her results to the study of certain algebraic varieties - certain toric varieties, ladder determinantal varieties and quiver varieties.Homogeneous spaces are fundamental objects in Geometry. Flag varieties constitute an important class of homogeneous spaces. Because of their rich geometry and combinatorics, flag varieties form interesting and important objects of study in the areas of Algebraic Geometry, Algebraic Groups, Representation Theory & Combinatorics. The Schubert varieties in the flag varieties provide a powerful machinery for the study of flag varieties.