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

海曼9701218海曼一直致力于一个项目的几年时间，以证明的插图加西亚和海曼给予组合模型麦克唐纳多项式，沿着一系列插图的海曼对角谐波，使用层同调方法适用于希尔伯特计划的点在平面和相关的代数品种。一旦完成，就有可能应用所得到的工具来寻求麦克唐纳多项式许多显着性质的统一几何解释。 这是计划研究的第一部分。 海曼和他的学生W.布罗克曼，寻求申请最近的工作Broer的某些方面的上述研究，发现了新的连接其他代数品种的一个和两个参数Kostka多项式。 这些发现导致了对拉斯科原子的几何解释，并产生了诱人的几何图形，其进一步的研究将构成计划研究的第二部分。 作为计划研究的第三部分和最后一部分，海曼将恢复他早期的工作Hecke代数和Kazhdan-Lusztig多项式，动机是他仍然未解决的Astratures Hecke代数字符，以及相关的问题，找到一个令人满意的组合解释Kazhdan-Lusztig多项式。 本研究关注组合学和代数几何之间的相互作用。组合学的目标之一是找到研究如何排列离散对象集合的有效方法。离散系统的行为对现代通信极为重要。例如，大型网络的设计，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，而且还使用分析，拓扑学和组合学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中得到应用。

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.