Topological Combinatorics of Posets, Totally Nonnegative Varieties and Crystals
偏序集、全非负簇和晶体的拓扑组合
基本信息
- 批准号:1200730
- 负责人:
- 金额:$ 15万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2012
- 资助国家:美国
- 起止时间:2012-08-01 至 2016-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
This project focuses on (1) stratified spaces coming from such areas as combinatorial representation theory and algebraic statistics; (2) the combinatorics of their closure posets; and (3) algebraic analogues with applications to geometric group theory, representation theory, and enumerative combinatorics. Building on the PI's past work studying the homeomorphism type of the totally nonnegative part of the unipotent radical of an algebraic group, the PI now will study the totally nonnegative part of the Grassmannian, in collaboration with Lauren Williams, with the long-range goal of determining the homeomorphism type of the totally nonnegative part of more general flag varieties; such topological analysis inevitably reveals a great deal of combinatorial and representation theoretic information in the process -- for example, to understand the nonnegative part of arbitrary flag varieties would very likely require a vast generalization of Postnikov's theory of reduced and nonreduced plabic graphs. Another focus of the project is on the development of new techniques and the streamlining of existing ones in poset topology, specifically building upon the PI's past work on discrete Morse theory for poset order complexes. The motivating application is to obtain, in collaboration with Cristian Lenart, a better understanding of the combinatorial structure of crystal graphs, guided by questions about their poset topology which will enrich and expand upon in new directions the local structure uncovered by Stembridge. Combinatorics is the mathematics of how to organize discrete data in ways that make it manageable to analyze. Topological combinatorics focuses on geometric data. For example, the set of solutions to a system of equations, such as one might encounter in an engineering problem, often can be split in a natural way into smaller pieces called cells that are much easier to understand. Topological combinatorics, the focus area of this project, can be used to understand how these cells fit together, focusing on finite data that can be used more easily in calculations. Specifically, partially ordered sets, or posets, are a combinatorial tool for describing incidences among these pieces. A long-term project of the PI is to develop efficient techniques for studying these partially ordered sets by a method called discrete Morse theory, which allows one to analyze the geometric object by building it over time, attaching its pieces in succession, and recording what happens at the moments in time when fundamental changes in the structure occur.
本项目重点研究(1)来自组合表示理论和代数统计等领域的分层空间;(2)闭包集的组合;(3)代数类似物在几何群论、表示论和枚举组合学中的应用。在PI过去研究代数群的单能根的完全非负部分的同胚类型的基础上,PI现在将与Lauren Williams合作,研究Grassmannian的完全非负部分,其长期目标是确定更一般的旗变体的完全非负部分的同胚类型;这种拓扑分析在这个过程中不可避免地揭示了大量的组合和表示理论信息——例如,要理解任意标志变体的非负部分,很可能需要对Postnikov的约简和非约简平面图理论进行广泛的推广。该项目的另一个重点是在偏序拓扑中开发新技术和简化现有技术,特别是建立在PI过去关于偏序复合体的离散莫尔斯理论的工作基础上。在与Cristian Lenart的合作中,这项激励人心的应用是为了更好地理解晶体图的组合结构,通过对其偏序集拓扑的问题进行指导,这将丰富和扩展Stembridge发现的局部结构的新方向。组合学是关于如何组织离散数据,使其易于分析的数学。拓扑组合学主要研究几何数据。例如,一个方程组的解的集合,比如在一个工程问题中可能遇到的,通常可以以一种自然的方式分成更小的块,称为单元,这样更容易理解。拓扑组合学,这个项目的重点领域,可以用来理解这些细胞是如何组合在一起的,专注于有限的数据,可以更容易地在计算中使用。具体地说,偏序集,或偏序集,是描述这些块之间的关联的组合工具。PI的一个长期项目是通过一种称为离散莫尔斯理论的方法来开发研究这些部分有序集合的有效技术,该方法允许人们通过随着时间的推移构建几何对象,连续连接其各个部分,并记录在结构发生基本变化的时刻发生的情况来分析几何对象。
项目成果
期刊论文数量(0)
专著数量(0)
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会议论文数量(0)
专利数量(0)
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Patricia Hersh其他文献
Chains of Modular Elements and Lattice Connectivity
- DOI:
10.1007/s11083-006-9053-x - 发表时间:
2007-01-24 - 期刊:
- 影响因子:0.300
- 作者:
Patricia Hersh;John Shareshian - 通讯作者:
John Shareshian
Lexicographic Shellability for Balanced Complexes
- DOI:
10.1023/a:1025044720847 - 发表时间:
2003-05-01 - 期刊:
- 影响因子:0.900
- 作者:
Patricia Hersh - 通讯作者:
Patricia Hersh
Patricia Hersh的其他文献
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{{ truncateString('Patricia Hersh', 18)}}的其他基金
Topological and Algebraic Combinatorics of Posets and Stratified Spaces
偏序集和分层空间的拓扑和代数组合
- 批准号:
1953931 - 财政年份:2020
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Topological and algebraic combinatorics of posets and stratified spaces
偏序集和分层空间的拓扑和代数组合
- 批准号:
1500987 - 财政年份:2015
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Algebraic and topological combinatorics
代数和拓扑组合数学
- 批准号:
1002636 - 财政年份:2009
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Algebraic and topological combinatorics
代数和拓扑组合数学
- 批准号:
0757935 - 财政年份:2008
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Algebraic and topological combinatorics of posets
偏序集的代数和拓扑组合
- 批准号:
0500638 - 财政年份:2005
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
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