I will describe the Chern-Schwartz-MacPherson (CSM) cycles of matroid, which are a collection of polyhedral fans introduced by López de Medrano, Rincón, and myself. When the matroid arises from a hyperplane arrangement, these fans encode the CSM class of the complement in its wonderful compactification. For arbitrary matroids these classes have found further applications. For instance the CSM cycles of matroids are an ingredient in Ardila, Denham, and Huh’s proof of Brylawski’s and Dawson’s conjectures and they also provide a Chow theoretic description of Speyer’s g-polynomial of a matroid arising from K-theory. Generalising the approach for matroids, I will explain how CSM classes can also be defined and used to study more general objects in tropical geometry.
This is based on joint work in progress with Lucia López de Medrano and Felipe Rincón.