New posts in Matroids

Are there matroids with $\mathcal F(M) \subsetneq \mathcal F(M^*)$ or $\mathcal C(M) \subsetneq \mathcal C(M^*)$?

Pairs of matroids $(M_1, M_2)$ with $\mathcal C(M_1) = \mathcal B(M_2)$

Applications of matroid theory.