Chern-Schwartz-MacPherson classes of matroids

Chern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes of the tangent bundle to singular and non-complete algebraic varieties. I will present a combinatorial analogue of CSM classes for matroids, motivated by the geometric case of hyperplane arrangements. The CSM classes of matroids are polyhedral fans which satisfy a balancing condition, in other words they are Minkowski weights. One goal for defining these classes is to express matroid invariants using the language of algebraic geometry and in turn use geometric intuition to study the properties of these invariants. The first example is the shifted characteristic polynomial of a matroid, which for graphical matroids is related to the chromatic polynomial. CSM classes can be used to study more general objects beyond the world of matroids, known as tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón.