RUI: Structural and enumerative problems on simplicial complexes

简单复合体是用来近似熟悉的几何空间的离散对象。它们根植于许多数学分支的历史发展，可以追溯到18世纪欧拉的工作。在过去的五十年里，几何组合学领域经历了巨大的发展。简单复合体的离散性质使它们非常适合于计算机实现，并且它们在数学生物学、优化、统计数据分析和计算机图形学等领域继续具有实际的现代应用。该项目还包括初高中学生的数学推广，以及本科生参与研究项目的承诺。这项资助将通过为学生提供技术培训和让他们体验从事原创科学研究的兴奋来支持这些努力。本研究的目的是进一步了解某些简单复合物族的拓扑结构和组合结构之间的相互作用。具体来说，我们试图理解某些条件，如图的可着色性或矩阵结构，如何影响某些简单复合物家族的组合结构。这些问题以组合几何为基础，但使用了交换代数、离散几何、代数拓扑和代数几何领域的工具，并且已经应用于这些领域。

Simplicial complexes are discrete objects that are used to approximate familiar geometric spaces. They are rooted in the historical development of many branches of mathematics, dating back to work of Euler in the 1700s. Over the past fifty years, the field of geometric combinatorics has experienced tremendous growth. The discrete nature of simplicial complexes makes them well-suited to computer implementations, and they continue to have practical modern applications in the fields of mathematical biology, optimization, statistical data analysis, and computer graphics. This project also encompasses mathematical outreach with middle and high school students, along with a commitment to involving undergraduate students in research projects. This grant will support these endeavors by providing students with technical training and exposing them to the excitement of engaging in original scientific research.The objective of the research is to further our understanding of the interplay between the topological and combinatorial structures of certain families of simplicial complexes. Specifically, we seek to understand how certain conditions, such as graph colorability or matroidal structures, affect the combinatorial structure of certain families of simplicial complexes. The problems are based in combinatorial geometry, but employ tools from, and have had applications to, the fields of commutative algebra, discrete geometry, algebraic topology, and algebraic geometry.