Some Algebraic Varieties Related to Flag Varieties

9971295本文讨论了齐次空间中一些重要的代数变种、Flag变种及其Schubert变种的若干问题。在与Musili和Seshadri的合作中，P.I.通过发展-现在众所周知的-标准单项式理论，在旗帜品种几何领域取得了重大突破。这一理论是对20世纪50年代发展起来的经典霍奇-杨理论的概括。这一理论导致了数量惊人的几何、表示理论和组合结果。利用她在标准单项式理论方面的工作，P.I.解决了舒伯特变种的许多重要问题——几何问题和表示论问题。最近，P.I.成功地将她的成果应用于某些代数品种的研究-某些环型品种，阶梯行列式品种和颤抖型品种。齐次空间是几何中的基本对象。标志变异构成了齐次空间的重要类别。由于其丰富的几何和组合性质，标志变异在代数几何、代数群、表示论和组合学等领域形成了有趣而重要的研究对象。旗类品种中的舒伯特品种为旗类品种的研究提供了强有力的机制。

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.