Topological and algebraic combinatorics of posets and stratified spaces

这个研究项目是在离散数学，即数学领域提供了理论基础的计算机科学，以及最近的生物学的一些实质性部分。 PI特别注重开发新的方法，将几何和拓扑技术与直觉结合起来。 近年来，PI特别专注于寻找有效的方法来研究满足自然产生的约束的实值矩阵空间上的拓扑组合结构，例如行列式和所有未成年人都是非负的矩阵。 这种空间出现在理论数学领域，如表示论，也出现在应用领域。 例如，它们对我们理解电网中电流和电压之间的关系起着重要作用。 更多的理论结果有时可以提供令人惊讶的强大的见解，这些应用。 该项目还包括一项研究，研究不同点的配置如何在空间中移动而不会相互碰撞，采用抽象的表征理论观点。 研究员亦会继续协助STEM管道的发展，透过培训组合学的研究生，以及举办工作坊和其他活动，以启发和培育下一代科学家的发展。具体项目包括：（1）分析映射至完全非负品种的同胚纤维类型;（2）利用偏序集拓扑和对称函数理论的混合方法，研究了与分格有关的位形空间的稳定性，（3）分析了分格上的组合拓扑结构 电网空间;和（4）发展的偏序集理论的方法，多面体直径界特别好类的多面体，从操作研究的复杂性问题，关于线性规划的动机。 其中许多项目是合作的。 这项工作建立在PI过去的研究拓扑组合，特别是在偏序集拓扑和结合思想的几何拓扑与组合研究组合拓扑结构的分层空间。

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.