Award Announcement - RUI: Structural and Enumerative Problems on Simplicial Complexes

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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.