Symmetry and structure of polyhedra and polytopes

The study of combinatorial and symmetry properties of polyhedra and polytopes has a long and storied history going all the way back to the classification in ancient Greece of the Platonic solids. Our primary objects of investigation will be the generalizations of convex polyhedra and polytopes known as “abstract polytopes”. The goal of this talk is to introduce a number of open problems in areas of active investigation in the theory of abstract polytopes. Of particular focus will be the interconnections between the ways in which group theory is used as a representation tool for combinatorial descriptions of discrete geometric objects, and the ways in which symmetry is leveraged in describing combinatorial structure. Along the way we will review some of the recent history of the development of the subject, with particular attention on the role that the most symmetric - “regular” - polytopes play in the theory.