Tits's Dream: Buildings Over F1 and Combinatorial Flag Varieties

The theme of the lecture is the notion of points over F1, the field with one element. Several heuristic computations led to certain expectations on the set of F1-points: for example the Euler characteristic of a smooth projective complex variety X should be equal to the number of F1-rational points of a suitable F1-model. While a literal interpretation of F1-points as morphisms from Spec F1 into X gives the wrong answers, we find a natural interpretation in terms of points over the Krasner hyperfield. This leads to a natural realization of Tits's proposed geometries over F1 as the "tip of the iceberg" inside Borovic-Gelfand-White's combinatorial flag varieties. As far as time allows, we explain how this phenomenon extends to an interpretation of structure constants of cluster algebras in terms of quiver matroids. Namely, the Caldero-Chapoton formula yields an explicit description of the cluster algebra of a tame quiver in terms of elementary expressions in the Euler characteristics of associated quiver Grassmannians. The aforementioned principle (which is proven for these instances) identifies the Euler characteristics with the number of certain quiver matroids, which are diagrams of matroids and strong maps.