Topological Combinatorics of Posets, Totally Nonnegative Varieties and Crystals

该项目的重点是（1）分层空间来自组合表示论和代数统计等领域;（2）其闭包偏序集的组合学;（3）代数类似物与几何群论，表示论和枚举组合学的应用。 在PI过去研究代数群的幂单根的完全非负部分的同胚类型的工作的基础上，PI现在将与Lauren威廉姆斯合作研究格拉斯曼的完全非负部分，其长期目标是确定更一般的旗簇的完全非负部分的同胚类型;这种拓扑分析不可避免地揭示了该过程中的大量组合和表示理论信息--例如，要理解任意旗型的非负部分，很可能需要对Postnikov的约化理论进行广泛的推广，非约平面图 该项目的另一个重点是在偏序集拓扑结构中开发新技术和简化现有技术，特别是建立在PI过去关于偏序集序复合体离散莫尔斯理论的工作基础上。 激励的应用程序是获得，与克里斯蒂安勒纳特，更好地了解组合结构的晶体图，指导问题，他们的偏序集拓扑结构，这将丰富和扩大后，在新的方向上的地方结构发现的Stembridge。 组合学是一门研究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，一个方程组的解集，比如一个工程问题中可能遇到的解，通常可以以一种自然的方式分成更小的部分，称为单元，更容易理解。 拓扑组合学是该项目的重点领域，可用于理解这些单元如何组合在一起，重点是可以更容易地用于计算的有限数据。 具体来说，偏序集（英语：partially ordered sets）是描述这些片段之间关联的组合工具。 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开发研究这些偏序集的有效技术，该理论允许人们通过随着时间的推移构建几何对象来分析几何对象，连续连接其片段，并记录结构发生根本变化时发生的时刻。

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.