Some Algebraic Varieties Related to Flag Varieties

9971295本文讨论了与齐性空间有关的一些重要代数簇、Flag簇及其Schubert簇的一些问题。 在与穆斯利和塞沙德里的合作中，P.I.在旗形几何学领域取得了重大突破，发展了现在众所周知的标准Monomial理论。 这个理论是20世纪50年代发展起来的经典的HODGE-YOUNG理论的推广。 这一理论导致了惊人数量的几何，表示理论和组合的后果。 利用她在标准单性理论方面的工作，P.I.解决了舒伯特簇的许多重要问题-几何和表示理论。最近，PI已成功地将她的结果应用于某些代数簇的研究-某些复曲面簇，阶梯行列式簇和复曲面簇。齐性空间是几何学的基本对象。 旗簇是一类重要的齐性空间。 由于其丰富的几何和组合学，标志品种形式有趣的和重要的研究对象，在代数几何，代数群，表示论组合数学领域。 旗形品种中的舒伯特品种为旗形品种的研究提供了强有力的工具。

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.