The higher Stasheff-Tamari posets and the higher Bruhat orders

The higher Bruhat orders are a generalization of weak Bruhat order on S_n. They can be defined combinatorially in terms of inversion sets, generalizing the notion of inversion set of a permutation, or geometically as sets of d-faces of an n-cube, generalizing the description of S_n as paths through an n-cube. The higher Stasheff-Tamari posets (which generalize the Tamari lattices) have an analogous geometric definition where the cube is replaced by a simplex. In this talk, I will review all the necessary definitions, and then discuss a new combinatorial "inversion set" description of the higher Stasheff-Tamari posets which amounts to giving from each Stasheff-Tamari poset into a corresponding higher Bruhat order.