Combinatorics and Algebraic Geometry -- Macdonald Polynomials, Hilbert Schemes, and Related Topics

组合学和代数几何——麦克唐纳多项式、希尔伯特方案和相关主题

• 批准号: 9701218
• 负责人: Mark Haiman
• 金额: $ 14.04万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9701218&HistoricalAwards=false
• 关键词: Combinatorics; Algebraic; Geometry; Macdonald; Polynomials

Haiman 9701218 Haiman has been working on a project of several years duration to prove conjectures of Garsia and Haiman giving a combinatorial model for Macdonald polynomials, along with a series of conjectures of Haiman on diagonal harmonics, using sheaf-cohomological methods applied to the Hilbert scheme of points in the plane and related algebraic varieties. Once that is done, it will be possible to apply the resulting tools to seek unified geometric explanations of the Macdonald polynomials' many remarkable properties. This constitutes the first part mf the planned research. Haiman and his student W. Brockman, seeking to apply recent work of Broer to certain aspects of the above study, have discovered new connections linking other algebraic varieties to the one- and two-parameter Kostka polynomials. These discoveries have led to a geometric explanation of Lascoux's atoms and to tantalizing conjectures whose further study will form the second part of the planned research. As a third and final part of the planned research, Haiman will resume his earlier work on Hecke algebras and Kazhdan-Lusztig polynomials, motivated by his still unsolved conjectures on Hecke algebra characters, and the related problem of finding a satisfactory combinatorial interpretation of Kazhdan-Lusztig polynomials. This research concerns the interplay between combinatorics and algebraic geometry. One of the goals of combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis, topology, and combinatorics, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

海曼9701218海曼从事一个持续数年的项目，以证明Garsia和Haiman的猜想为MacDonald多项式提供了组合模型，并使用Haiman On On On On On On on dogonal On on the Hileap-Onalical方法应用了Hileaf-Ongomological Spine，平面中的点和相关的代数品种。完成此操作后，就可以应用所得工具来寻求麦克唐纳多项式的许多显着特性的统一几何解释。 这构成了计划研究的第一部分。 海曼和他的学生W. Brockman试图将兄弟的最新作品应用于上述研究的某些方面，他们发现了将其他代数品种与单参数Kostka多项式相关联的新联系。 这些发现导致了对拉斯科克斯原子的几何解释，并引起了进一步研究将构成计划研究的第二部分的猜想。 作为计划研究的第三部也是最后一部分，海曼将恢复他对Hecke代数和Kazhdan-Lusztig多项式的早期工作，这是因为他对Hecke代数角色的仍未解决的猜想以及发现令人满意的kazhdan-atorial解释的相关问题Lusztig多项式。 这项研究涉及组合学与代数几何形状之间的相互作用。组合主义者的目标之一是找到有效的方法来研究如何安排对象的离散收集。离散系统的行为对于现代通信非常重要。例如，大型网络的设计，例如在电话系统中发生的网络，以及计算机科学中算法的设计与离散对象集有关，这利用了组合研究。代数几何形状是现代数学最古老的部分之一，但在过去的25年中，它具有革命性的开花。它起源于它处理的数字可以通过最简单的方程（即多项式）在平面中定义的数字。如今，该领域不仅利用了代数，而且从分析，拓扑和组合学中使用方法，相反，在这些领域以及物理学，理论计算机科学和机器人技术中都可以找到应用。