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Mullins, Patrick

Mathematics and Statistics

2020

M.S.

A matroid abstracts the notions of dependence common to linear algebra, graph theory, and geometry. We show the equivalence of some of the various axiom systems which define a matroid and examine the concepts of matroid minors and duality before moving on to those matroids which can be represented by a matrix over any field, known as regular matroids. Placing an orientation on a regular matroid allows us to define certain lattices (discrete groups) associated to the matroid. These allow us to construct the Jacobian group of a regular matroid analogous to the Jacobian group of a graph. We then survey some recent work characterizing the matroid Jacobian. Finally we extend some results due to Eppstein concerning the Jacobian group of a graph to the case of regular matroids.