Mathematics Colloquia and Seminars

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One of the natural objects associated to every realisable matroid is a torus orbit on the Grassmannian. This underlies many of the appearences of matroid base polytope decompositions. We begin with some remarks independent of the geometry on so-called valuative functions of matroids, those functions which behave well in polytope decompositions. We then introduce the geometry, and turn the K-theory of the torus orbits on the Grassmannian into an entirely combinatorial picture of polyhedra and lattice points. We use this to bring non-realisable matroids into the picture, and to give related interpretations of two matroid invariants, namely the Tutte polynomial and an invariant of Speyer's giving combinatorial bounds on linear spaces in tropical geometry. This talk is based on joint work with David Speyer.