The first mathematicians that mentioned the desire for a field F_1 with one element, and of geometry over such an elusive object, was Jacques Tits who pursued this idea in the 1950s. His hope was to explain the analogy between geometries over finite fields F_q and certain incidence geometries that behave like the limit q -> 1. Much later, around 2000, Borovic, Gelfand and White expanded Tits's perspective towards combinatorial flag varieties, which are incidence geometries that stem from matroid theory.

In this talk, we introduce a formalism for algebraic geometry over F_1 that captures all these effects in terms of moduli spaces of flag matroids: finite field geometries emerge as F_q-rational points of these moduli spaces, combinatorial flag varieties arise as rational points with values in the so-called Krasner hyperfield and the Tits's incidence geometries resurface as the subsets of closed points of these moduli spaces. We conclude the talk with an explanation on how algebraic groups generalize to F_1 and in which sense SL(n) acts on the moduli space of flags.