I will talk about the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B'(Kn) of the graphical matroid of the complete graph Kn is homeomorphic to the space of phylogenetic trees Tn .
This is joint work with Federico Ardila.