Barry Monson - Realizations of regular abstract polytopes

A regular abstract polytope is a poset having the `essential' structural features of the face lattice of a regular convex polytope, including transitivity of on flags. Other examples include regular tessellations and honeycombs, star-polyhedra and many less familiar beasts. Since need not actually be a lattice or have a particularly nice geometric realization, it is interesting to describe all realizations for . The basic theory of realizations was developed by McMullen (1989), who used geometric methods to describe a crucial family of real representations for .

After an explicit, but brief, description of these objects, I will discuss work with Asia Ivic Weiss, in which we construct families of regular polytopes parametrized by certain rings of algebraic integers. Typically, our understanding of the realization cones for these polytopes is very limited.