Computational/Combinatorial Considerations In Topology, Coxeter Groups, and Representation Theory

主要研究者：Billey， Sara提案编号：DMS - 0800978机构：华盛顿大学标题：拓扑、Coxeter群和表示理论中的计算/组合考虑本提案概述了一个雄心勃勃的研究计划，主要研究三个领域的问题：仿射格拉斯曼理论、排列基础和受复杂性理论启发的表示理论。中心主题是通过研究非常具体的偏序集、Weyl群、生成函数和控制微观层次结构的分区来促进这些领域的计算和理解，而我们通常在第一次瞥见这些主题时忽略了这些结构。代数组合学的核心理念是，通过确定关键的组合结构，数学的各个方面都可以变得更精确、更具体、更计算可行。借用分析中的一个术语，这个建议是关于“微观局部数学”的。所有提出的工作将在纯数学和理论计算机科学的几个领域产生广泛的影响。所有提议的工作都将以计算为重点，这将进一步发展计算机证明技术。所有提议的工作都将涉及人类影响。PI在指导各级学生和初级教师方面有着良好的记录。这笔拨款将大大加强华盛顿大学组合学小组的垂直整合研究环境，并为该小组提供解决这些难题所需的资源。

ABSTRACTPrincipal Investigator: Billey, Sara Proposal Number: DMS - 0800978Institution: University of WashingtonTitle: Computational/Combinatorial Considerations In Topology, Coxeter Groups, and Representation TheoryThis proposal outlines an ambitious research program attacking problems in three areas: the affine Grassmannian, fundamentals of permutations, and representation theory inspired by complexity theory. The central theme is to facilitate computation and understanding in these areas through study of very specific posets, Weyl groups, generating functions, and partitions controlling the micro level structures which we often overlook on the first glimpse of the subjects.At the heart of algebraic combinatorics is the philosophy that every aspect of mathematics can be made more precise, more concrete and more computationally feasible by identifying key combinatorial structures. Borrowing a term from analysis, this proposal is about ``micro local mathematics''. All of the work proposed will have a broad impact in several areas of pure math and theoretical computer science. All of the proposed work will have a computational focus which will further develop computer proof techniques. All of the proposed work will have a human impact component. The PI has a strong track record of mentoring students at all levels and junior faculty. This grant will greatly enhance the vertically integrated research environment in the Combinatorics Group at the University of Washington and give the group the resources needed to attack these hard problems.