Topological Combinatorics of Posets, Totally Nonnegative Varieties and Crystals

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.