Algebraic combinatorics: symmetric orbit closures and Schubert calculus

该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，其应用包括最优化、计算机科学和统计物理。这项研究的中心焦点是对出现在离散数学、几何和对称学研究界面上的多项式方程的研究。该项目的一个关键组成部分将是对学生(高中、本科生和研究生)和博士后教师的培训。因此，我们希望帮助加强美国的STEM教育和研究基础设施。该研究人员的早期工作已导致等变上同调与厄米特矩阵的谱、具有Kazhdan-Lusztig多项式的组合交换代数、具有随机字中最长递增子序列的组合K-理论以及具有文献计量指标的划分组合之间的精确联系。这样的例子促使人们找到更广泛、更统一的组合律，这也是本项目的目标。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.