Families of matroids have appeared in different disguises during the last few decades: combinatorial flag varieties arise from flag matroids, Macphersonians appear as spaces of oriented matroids, Dressians consist of valuated matroids. With the advent of F1-geometry, we are able to understand these spaces through an algebro-geometric perspective as rational point sets of moduli spaces of (flag) matroids and place them into a larger landscape of geometric objects stemming from combinatorics.
In this talk, I will review my joint works with Matthew Baker and with Manoel Jarra on the topic. This includes a review of Baker-Bowler theory, a primer to F1-geometry and the construction of the moduli spaces of (flag) matroids. We explain how to recover the aforementioned geometric objects and comment on applications and future directions.