Georgia Institute of Technology College of Sciences

The Jacobian (or sandpile group) of a graph is a well-studied group associated with the graph, known to biject with the set of spanning trees of the graph via a number of classical combinatorial mappings. The algebraic definition of a Jacobian extends to regular matroids, but without the notion of vertices, many such combinatorial bijections fail to generalize. In this talk, I will discuss how orientations provide a way to overcome such obstacle. We give a novel, effectively computable bijection scheme between the Jacobian and the set of bases of a regular matroid, in which polyhedral geometry plays an important role; along the way we also obtain new enumerative results related to the Tutte polynomial. This is joint work with Spencer Backman and Matt Baker.