Georgia Institute of Technology College of Sciences

A matroid is modular if its lattice of flats is self-dual. Prototypical examples include the Boolean matroid $U_{n,n}$ and finite projective geometry matroid $M(\mathbb{F}_q^n)$. We discovered that the space of quotients of any modular matroids has remarkable structure. For instance, quotients of the Boolean matroid $U_{n,n}$, also known as the set of all matroids on $[n]$, are equipped with operations such as matroid duality and matroid intersection. All these operations exist canonically for quotients of arbitrary modular matroids. We will showcase some implications of these operations, including a generalization of stable intersection on the Bergman fan of any modular matroid.