Topological Combinatorics of Posets, Totally Nonnegative Varieties and Crystals

偏序集、全非负簇和晶体的拓扑组合

• 批准号: 1200730
• 负责人: Patricia Hersh
• 金额: $ 15万
• 依托单位: North Carolina State University
• 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: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200730&HistoricalAwards=false
• 关键词: Topological; Combinatorics; Posets; Totally; Nonnegative

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.

该项目着重于（1）来自组合代表理论和代数统计等领域的分层空间； （2）闭合姿势的组合； （3）代数类似物与几何群体理论，表示理论和列举组合学的应用。 基于PI过去的工作，研究了代数群体的完全无负部分的同构类型的同态类型，PI现在将研究与劳伦·威廉姆斯（Lauren Williams）合作的完全无负的部分，其长期目标是确定完全非象征性的一般旗帜的同源性类型的长期目标，这种拓扑分析不可避免地揭示了此过程中的大量组合和表示理论信息 - 例如，要了解任意标志品种的非负部分，很可能需要对后尼科夫的减少和非还原性牙齿图的理论进行广泛的概括。 该项目的另一个重点是新技术的开发以及Poset拓扑中现有的技术的精简，特别是基于PI过去的莫尔斯（Morse）莫尔斯（Morse）理论的POSET订单复合物的工作。 激励的应用是与克里斯蒂安·莱纳特（Cristian Lenart）合作获得对晶体图的组合结构的更好理解，并以有关其POSET拓扑的问题为指导，这些问题将富集和扩展在新的方向上，由Stembridge发现的局部结构。 组合学是如何以使其可以进行分析的方式组织离散数据的数学。 拓扑组合学的重点是几何数据。 例如，一组方程系统的解决方案，例如在工程问题中可能会遇到的，通常可以自然地将其分为称为单元格的较小细胞，这些细胞更容易理解。 拓扑组合学是该项目的焦点区域，可用于了解这些细胞如何融合在一起，重点是在计算中更容易使用的有限数据。 具体而言，部分有序的集合或posets是用于描述这些部分中发生率的组合工具。 PI的一个长期项目是通过一种称为离散的Morse理论的方法来开发有效的技术来研究这些部分有序的集合，该方法使人们能够通过构建它随时间构建，连续地将其构建，并记录结构中基本变化时及时在及时发生的时刻进行分析。