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

Haiman 9701218 Haiman一直致力于一个持续数年的项目，以证明Garsia和Haiman的猜想，给出了Macdonald多项式的一个组合模型，以及Haiman关于对角调和的一系列猜想，使用应用于平面上点的Hilbert方案和相关代数簇的层上同调方法。一旦完成，就有可能应用所产生的工具来寻求对麦克唐纳多项式的许多显著性质的统一的几何解释。这是计划研究的第一部分。Haiman和他的学生W.Brockman试图将Broer最近的工作应用于上述研究的某些方面，他们发现了将其他代数变体与单参数和双参数Kostka多项式联系起来的新联系。这些发现导致了对LasCoux原子的几何解释，并产生了诱人的猜想，这些猜想的进一步研究将构成计划中的研究的第二部分。作为计划研究的第三部分，也是最后一部分，Haiman将继续他早期关于Hecke代数和Kazhdan-Lusztig多项式的工作，动机是他关于Hecke代数特征标的仍未解决的猜想，以及寻找Kazhdan-Lusztig多项式的满意组合解释的相关问题。这项研究涉及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有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析、拓扑学和组合学的方法，反过来，也在这些领域以及物理、理论计算机科学和机器人学中找到应用。

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.