A pivot rule is the mechanism that tells the simplex algorithm which path to take on a linear program from a given vertex to an optimal one. Together with Black, De Loera, and Lütjeharms, we introduced pivot polytopes as a mean to capture the behaviour of certain classes of pivot rules on a given linear program. While this gives a new perspective on pivot rules, it turns out that pivot polytopes are also of interest to combinatorialists. For instance, we showed that pivot rule polytopes relate flag matroid polytopes to nestohedra. In this talk, I will discuss how pivot polytopes of "boring" polytopes yield quite some amazing combinatorics, including associahedra, multiplihedra, and more general associative structures.
Hyperplanes can be used to describe polytopes, (oriented) matroids, Coxeter groups, and other combinatorial objects. Collections of hyperplanes dissect the ambient vector space into connected components called regions, and Varchenko-Gelfand introduced a ring from the regions of a real arrangement by considering all maps from the set of regions to the integers, with pointwise addition and multiplication. Varchenko-Gelfand gave an interesting presentation for this ring as (a quotient of) a polynomial ring, and an interesting interpretation of this presentation is given by Moseley. In this talk, we consider the intersection of a hyperplane arrangement with an open, convex set and recover analogues both of Varchenko-Gelfand's presentation and of Moseley's topological interpretation.
