Topological Combinatorics of Posets, Totally Nonnegative Varieties and Crystals

这个项目主要集中在(1)来自组合表示理论和代数统计等领域的分层空间；(2)它们的闭合偏序集的组合学；(3)代数类似及其在几何群论、表示论和计数组合学中的应用。在PI过去研究代数群的酉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Grassmanian的完全非负部分，长期目标是确定更一般的旗子变种的完全非负部分的同态类型；这种拓扑分析不可避免地在这个过程中揭示了大量的组合和表示理论信息--例如，要理解任意旗子变种的非负部分很可能需要对波斯尼科夫的约化和非约化积图理论进行广泛的推广。该项目的另一个重点是偏序集拓扑学中新技术的发展和现有技术的简化，特别是建立在PI过去关于偏序集序复形的离散Morse理论的工作基础上。激发灵感的应用是，与克里斯蒂安·勒纳特合作，在偏序集拓扑问题的指导下，更好地理解晶图的组合结构，这将在新的方向上丰富和扩展Stembridge发现的局部结构。组合学是关于如何以便于分析的方式组织离散数据的数学。拓扑组合学主要研究几何数据。例如，一组方程的解，就像在工程问题中可能遇到的那样，通常可以以一种自然的方式被分成更小的单元，这些单元更容易理解。拓扑组合学是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，重点是有限的数据，这些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部分之间的关联度的组合工具。PI的一个长期项目是开发一种名为离散莫尔斯理论的方法来研究这些偏序集的有效技术，该方法允许人们通过随着时间的推移构建几何对象，连续地附加其碎片，并记录结构发生根本变化的时刻发生的事情来分析几何对象。

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.