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

• 批准号: 1500987
• 负责人: Patricia Hersh
• 金额: $ 22万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500987&HistoricalAwards=false
• 关键词: Topological; algebraic; combinatorics; posets; stratified

This research project is in discrete mathematics, namely, the area of mathematics which provides the theoretical underpinnings for computer science as well as more recently for some substantial parts of biology. The PI particularly focuses on developing novel ways of combining geometric and topological techniques and intuition with combinatorial methods. In recent years, the PI has become particularly focused on finding effective ways to study topological-combinatorial structures on spaces of real-valued matrices satisfying naturally arising constraints, for instance matrices in which the determinant as well as all minors are nonnegative. Such spaces arise both in areas of theoretical mathematics such as representation theory and also in applications areas. For instance, they play an important role to our understanding of the relationship between current and voltage in electrical networks. The more theoretical results can sometimes give surprisingly powerful insights into such applications. The project also includes a study of how configurations of distinct points may move around in space without bumping into each other, taking an abstract, representation theoretic perspective. The PI will also continue her work in helping develop the STEM pipeline both through the training of graduate students in combinatorics and also through organizing workshops and other activities to help inspire and foster the development of the next generation of scientists.The specific projects include: (1) analysis of the homeomorphism type of fibers of maps to totally nonnegative varieties; (2) stability properties for configuration spaces related to the partition lattice via a mixture of poset topology and symmetric function theory; (3) analysis of combinatorial topological structure on spaces of electrical networks; and (4) development of poset-theoretic approaches to polytope diameter bounds for particularly nice classes of polytopes, motivated by complexity questions from operations research regarding linear programming. Many of these projects are collaborative. This work builds upon the PI's past research in topological combinatorics, and particularly in poset topology and in combining ideas of geometric topology with those of combinatorics to study combinatorial topological structure of stratified spaces.

该研究项目具有离散的数学，即数学领域，该领域为计算机科学提供了理论基础，以及最近为生物学的某些部分提供了基础。 PI尤其着重于开发新的方法，将几何和拓扑技术和直觉与组合方法结合在一起。 近年来，PI尤其专注于寻找有效的方法来研究满足自然产生约束的实价矩阵的空间的拓扑组合结构，例如，确定性和所有未成年人都是非负责人的矩阵。 这种空间在理论数学的领域（例如表示理论）以及应用领域都出现。 例如，它们对我们对电网中电流和电压之间关系的理解发挥了重要作用。 理论上的结果越多，有时可以对此类应用产生令人惊讶的强大见解。 该项目还包括对不同点的配置如何在太空中四处移动而不互相碰到的研究，采用抽象，表示理论的观点。 PI还将继续她的工作，以通过培训组合学的研究生，以及组织研讨会和其他活动来帮助发展STEM管道，以帮助启发和促进下一代科学家的发展。（1）分析：（1）分析地图纤维类型的同型纤维类型，以使其完全非对型品种; （2）通过Poset拓扑和对称函数理论的混合物与分区晶格有关的配置空间的稳定性属性； （3）电网空间上组合拓扑结构的分析； （4）开发Poset理论方法，用于多型直径的界限，以供多个类型的多种类型，这是由有关线性编程的操作研究中的复杂性问题所激发的。 这些项目中的许多都是协作的。 这项工作是基于PI过去在拓扑组合学方面的研究，尤其是在Poset拓扑结构以及将几何拓扑结合在一起的思想与组合学的思想相结合，以研究分层空间的组合拓扑结构。