Algebraic combinatorics: symmetric orbit closures and Schubert calculus

这个项目的目标是构建组合模型。组合问题出现在数学的许多领域，并有应用，包括优化，计算机科学和统计物理。这项研究的中心焦点是研究出现在离散数学，几何和对称性研究的界面上的多项式方程。该项目的一个关键组成部分将是学生（高中，本科和研究生）和博士后教师的培训。因此，我们希望帮助加强STEM教育和研究基础设施在美国。早期的工作的研究者已经导致了精确的连接等变上同调与谱的Hermitian矩阵，组合交换代数与Kazhdan-Lusztig多项式，组合K-理论与最长的增加随机词的连续性，和分区组合学与文献计量指标。这些例子激发了寻找更广泛和更统一的组合定律，这是本项目的目标。这项研究将加深我们对这种联系的理解，并有助于发现新的关系。

The goal of this project is to construct combinatorial models. Combinatorial problems arise in many areas of mathematics and have applications that include optimization, computer science, and statistical physics. The central focus of this research is the study of polynomial equations that appear at the interface of the study of discrete mathematics, geometry, and symmetries. A key component of this project will be the training of students (high school, undergraduate, and graduate) and postdoctoral faculty. Thereby, we wish to help strengthen the STEM education and research infrastructure in the United States.Earlier work of the investigator has led to precise connections of equivariant cohomology with spectra of Hermitian matrices, combinatorial commutative algebra with Kazhdan-Lusztig polynomials, combinatorial K-theory with longest increasing subsequences in random words, and partition combinatorics with bibliometric indicators. Such examples motivate finding broader and more unified combinatorial laws, which is the goal of this project. This research will both deepen our understanding of such connections and help discover novel relationships.