We introduce a new presentation of the Chow ring of a matroid whose variables now admit a combinatorial interpretation via the theory of matroid quotients and display a geometric behavior analogous to that of nef classes on smooth projective varieties. We discuss various applications, including the recovery of the Hodge theory of matroids as developed by Adiprasito, Huh, and Katz. This is joint work with Spencer Backman and Connor Simpson. Speaker(s): Christopher Eur (UC Berkeley)