In joint work with Federico Ardila and June Huh, we introduce the conormal fan of a matroid, which is an analogue of the Bergman fan. We use it to give a Lagrangian interpretation of the Chern-Schwartz-MacPherson cycle of a matroid. We also develop tools for tropical Hodge theory to show that the conormal fan satisfies Poincaré duality, the Hard Lefschetz property, and the Hodge--Riemann relations. Together, these imply conjectures of Brylawski and Dawson about the log-concavity of the h-vectors of the broken circuit complex and independence complex of a matroid.