I'll describe a new invariant of matroids that takes the form of a
quasisymmetric function, then describe some of its pleasant properties.
One is that it defines a morphism between two known combinatorial Hopf
algebras. Another is that it helps answer a subtle question that arises
in work of Kapranov, Lafforgue, Hacking, Keel, Tevelev and others:
when does a matroid polytope decompose into smaller matroid polytopes?
Knowledge of quasisymmetric functions will not be assumed.

This is joint work with Lou Billera and Ning Jia.