Abstract: This talk is about a concrete convex geometry problem related to the combinatorics of affine Schubert varieties X(w). I will introduce some basic notions with pictures and examples. I will introduce the notions of unimodality and top-heaviness for arbitrary sequences. Then I will explore these notions for the sequences b_i^w of Betti numbers corresponding to the varieties X(w).

These sequences are always top-heavy, but not always unimodal. In the case of dominant elements in the affine Weyl group, the behavior of the corresponding sequences b_i^w is described by a convex polyhedron that only depends on a dominant weight and the Cartan matrix of the corresponding (finite) root system. I will measure how top-heavy some of these sequences are, and state some results and conjectures. Examples in ranks 2, 3, and higher will be given.