Matroid Hodge theory has been used for several spectacular recent resolutions of combinatorial conjectures. A good deal of the computations can be done in the toric variety of the permutahedron.

Berget, Eur, Spink and Tseng have developed tools to easily pass between Chow and K-theory computations in this setting, including introducing some tautological vector bundles for matroids. I'll talk about this as well as ongoing joint work of mine with Eur, Larson and Spink which extends it to delta-matroids, their Coxeter type BC counterpart.