Abstract: I shall continue giving a survey of current efforts to bring these two subjects together in the context of finite spectral triples.

Abstract: The associahedron is a well-known polytope, defined by Stasheff in 1963 as a cell complex, and given its first polytopal realizations by Haiman and by Lee in the 1980s. The past twenty years have seen ongoing interest in associahedra and their generalizations because of their connection to cluster algebras; the 1-skeleton of an associahedron is the exchange graph of a type A cluster algebra. More recently, associahedra have attracted attention in physics: Arkani-Hamed, Bai, He, and Yan showed that a particular realization of the associahedron encodes the scattering amplitudes for a particular quantum field theory. It turns out that the most natural way to construct the associahedron which the physicists needed is via the theory of quiver representations. I will explain this approach, and also how it can be generalized to obtain many other polytopes, some of which are also of interest to physicists. No knowledge of quiver representations is assumed; I will explain what I need.