Pivots, polytopes, and associative structures

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.