RUI: Structural and enumerative problems on simplicial complexes

单纯形复形是离散对象，用于近似熟悉的几何空间。它们植根于数学许多分支的历史发展，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增长。 单纯复形的离散性质使它们非常适合计算机实现，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领域中继续具有实际的现代应用。该项目还包括与初中和高中学生的数学推广，沿着致力于让本科生参与研究项目。 这项拨款将支持这些努力，为学生提供技术培训，并让他们接触到从事原创性科学研究的兴奋。研究的目的是进一步了解某些单纯复形家族的拓扑和组合结构之间的相互作用。 具体来说，我们试图了解某些条件，如图着色或拟阵结构，影响某些家庭的单纯复形的组合结构。 这些问题是基于组合几何，但采用工具，并已应用到交换代数，离散几何，代数拓扑和代数几何领域。

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.