Postnikov has shown that generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, have remarkable formulas for their volumes and lattice point counts. Additionally, generalized permutohedra can always be decomposed into certain easy-to-understand polytopes. I will discuss a generalization of these results to signed generalized permutohedra, polytopes whose edges are parallel to vectors of the form e_i - e_j, e_i + e_j, or e_i. Joint work with Chris Eur, Alex Fink, and Hunter Spink.

Cross-ratio degrees are intersection numbers on M_{0,n}-bar indexed by certain bipartite graphs. I'll discuss several angles on cross-ratio degrees, and give a combinatorial upper bound involving perfect matchings, with a geometric proof. Time permitting, I'll give a connection to tropical intersection theory.

By Mnëv's universality theorem, every singularity type appears in the realization spaces of a rank 3 matroid. While the proofs of this theorem are constructive, the resulting matroids have large ground sets even for the simplest singularities. I will present recent joint work with Dante Luber (TU Berlin) where we show that, over the complex numbers, the realization spaces of rank 3 matroids on ≤ 11 elements are all smooth, but there are singular realization spaces for matroids on ≥ 12 elements. We use this result to show that the open Grassmannian is not schön in the sense of Tevelev.