Vanishing theorems for combinatorial geometries

Matroids are combinatorial abstractions of linear independence.  The landmark development of Hodge theory of matroids showed intersection theoretic positivity for matroids.  In algebraic geometry, a counterpart of such Chow-theoretic intersection theory is K-theory.  We establish K-theoretic positivity for matroids.  Our results in particular give a new proof of the 20-year-old f-vector conjecture of Speyer.  Joint work with Alex Fink and Matt Larson.