Mathematical Sciences: Combinatorial Methods in Algebra and Geometry; Macdonald Polynomials, Diagonal Harmonics, and the Hilbert Scheme

9400934海曼这个项目涉及到麦克唐纳多项式相关模块和对角谐波模块的研究。这项工作有望在这些对象与代数组合学和代数几何的其他部分之间建立新的关系。初步工作已经建立了连接的框架，并产生了重要的新代数恒等式。这项研究属于组合学的广泛范畴，这是当今数学中最活跃的领域之一。从根本上说，组合学是对系统计数的研究。当对象难以列出时，计数可能会非常困难，组合学家会寻找克服这些困难的通用方法。今天的组合学利用了各种最先进和现代的数学技术。虽然它的根源可以追溯到几个世纪以前，但在过去的几十年里，这个领域有了爆炸性的发展。这种增长来自于它在通信和信息技术中的重要性，以及现代技术在计数问题上的成功。

9400934 Haiman This project involves research into modules associated with the Macdonald polynomials and the module of diagonal harmonics. The work promises to establish new relationships between these objects and other parts of algebraic combinatorics and algebraic geometry. Preliminary work has already established the framework for the connections and has produced important new algebraic identities. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. At its roots, combinatorics is the study of systematic counting. Counting can be incredibly difficult when the objects are difficult to list, and combinatorists look for general methods for overcoming these difficulties. Today's combinatorics makes use of a wide variety of the most advanced and modern mathematical techniques. Although its roots go back several centuries, the field has had an explosive development in the past few decades. This growth comes from its importance in communications and information technology and from the success of modern techniques to problems of counting.