Intersection theory studies how subvarieties of an algebraic variety X intersect. Algebraically, this information is encoded in the Chow ring A(X). When X is the toric variety of a simplicial fan, Brion gave a presentation of A(X) in terms of generators and relations, and Fulton and Sturmfels gave a "fan displacement rule” to intersect classes in A(X), which holds more generally in tropical intersection theory. In these settings, intersection theoretic questions translate to algebraic combinatorial computations in one point of view, or to polyhedral combinatorial questions in the other. Both of these paths lead to interesting combinatorial problems, and in some cases, they are important ingredients in the proofs of long-standing conjectural inequalities. This talk will survey three problems on matroids and root systems that arise in combinatorial intersection theory. It will feature joint work with Montse Cordero, Graham Denham, Chris Eur, June Huh, Carly Klivans, and Raúl Penaguião. I will make the talk as accessible as I am able to.