Higher-Dimensional Sandpile Groups and Matrix-Tree MultijectionsMcDonough, AlexcreatorKlivans, CarolineAdvisorChan, MelodyReaderMartin, JeremyReaderBrown University. Department of Mathematicssponsor2021xi, 173 p.born digitalThesis (Ph. D.)--Brown University, 2021thesesTraditionally, the sandpile group is defined on a graph and the matrix-tree theorem says that this group's size is equal to the number of spanning trees. There are extensions of the matrix-tree theorem which express the size of a generalized sandpile group as a weighted count of generalized spanning trees. In this manuscript, we elevate this enumerative relationship to a combinatorial relationship by constructing a family of many-to-one maps, which we call multijections. This construction is geometric and involves a periodic tiling of n-space made up of non-convex polyhedra.Combinatorial geometrySandpile groupTiling (Mathematics)EnglishRPB20210607dissertations