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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

基本信息

批准号：
1953931
负责人：
Patricia Hersh
金额：
$ 15万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953931&HistoricalAwards=false
关键词：
Topological Algebraic Combinatorics Posets Stratified

项目摘要

This project is in combinatorics, namely the area of math that provides the theoretical underpinnings for computer science and also arises naturally in other areas such as the study of DNA and RNA mutation in biology. This project in particular aims to develop new ways to combine combinatorial techniques with methods in topology. There is a rich tradition of such interactions that really began to flourish starting in the 1970's, the starting point being the observation that counting by inclusion-exclusion (namely counting things up via Venn diagrams and related enrichments) may be accomplished topologically. The PI has begun bringing more geometric topological tools to the mix in addressing offshoot questions about rather mysterious spaces, including spaces of real-valued matrices. Her techniques involve breaking the spaces into more manageable pieces and analyzing structures on these spaces called partially ordered sets (posets) for organizing the pieces, using geometric ideas to study how pieces glue together. The new work will focus on spaces arising from electrical networks, spaces coming from totally nonnegative matrices (i.e. matrices of real numbers where all subdeterminants are nonnegative), generalizations of these, and other related stratified spaces which seem to share important features also making them seem likely amenable to somewhat the same type of analysis. In addition to carrying out this research, the PI also will continue her efforts to encourage and train young mathematicians, including members of underrepresented groups, and to disseminate her results.Expressed in somewhat more technical terms, the PI will study the combinatorial and topological structure of partially ordered sets and of stratified spaces related to electrical networks, topological and combinatorial questions related to the Bruhat graph and generalizations of this, electrical analogues of subword complexes and Kazhdan-Lusztig polynomials, subword complexes and their interior dual block complexes and generalizations of these, fibers of important maps seemingly sharing with each other key structures encapsulated in part via the structure of subword complexes and generalizations thereof, as well as questions about polytope diameter and complexity of the simplex algorithm for linear programming approached via poset-theoretic methods. Other methods to be used include shellability, the application of poset-theoretic results of Dyer, Brenti and others as well as potential new extensions of these, methods the PI began to develop in her work on regular cell complexes in total positivity, new definitions suggested by the PI which she believes give helpful new perspective, and the PI's new techniques for combining poset-theoretic methods with tools of discrete geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目是在组合学，即数学领域，为计算机科学提供理论基础，也自然出现在其他领域，如生物学中的DNA和RNA突变的研究。该项目特别旨在开发新的方法，将联合收割机组合技术与拓扑学方法相结合。这种相互作用有一个丰富的传统，从20世纪70年代开始真正开始蓬勃发展，其出发点是观察到通过包含-排除（即通过维恩图和相关的丰富来计数）可以拓扑地完成。 PI已经开始将更多的几何拓扑工具用于解决关于相当神秘的空间（包括实值矩阵空间）的分支问题。她的技术包括将空间分解为更易于管理的片段，并分析这些空间上的结构，称为偏序集（posets），用于组织片段，使用几何思想来研究片段如何粘合在一起。新的工作将集中在空间所产生的电气网络，空间来自完全非负矩阵（即矩阵的真实的号码，其中所有的子行列式是非负的），推广这些，以及其他相关的分层空间，似乎共享重要的功能，也使他们似乎可能服从某种相同类型的分析。除了开展这项研究，PI还将继续努力鼓励和培训年轻的数学家，包括代表性不足的群体的成员，并传播她的成果。用更专业的术语来表达，PI将研究偏序集和与电网络相关的分层空间的组合和拓扑结构，与Bruhat图及其推广有关的拓扑和组合问题，子字复形和Kazhdan-Lusztig多项式的电模拟，子字复形及其内部对偶块复形及其推广，纤维的重要地图似乎共享彼此的关键结构封装在部分通过结构的子字复合体及其推广，以及问题的多面体直径和复杂性的单纯形算法的线性规划接近通过偏序集理论的方法。其他要使用的方法包括可壳性，Dyer，Brenti和其他人的偏序集理论结果的应用以及这些结果的潜在新扩展，PI开始在她关于总阳性的规则细胞复合体的工作中开发的方法，PI建议的新定义，她认为这些新定义提供了有用的新视角，PI的新技术结合偏序集该奖项反映了NSF的法定使命，并通过使用基金会的智力价值进行评估，更广泛的影响审查标准。