Face numbers: the upper bound side of the story

Abstract

We survey the theory of face numbers of simplicial complexes through the lens of upper bound type results and neighborliness. We focus on the classes of polytopes, simplicial spheres, and simplicial manifolds, along with the classes of centrally symmetric polytopes and centrally symmetric simplicial spheres. Along the way, we sketch some of the ideas and methods used in the proofs. We also highlight some of the many open problems in the field.