The f-vector of a simplicial complex X is the vector whose i-th component is the number of i-dimensional faces of X. The problem of characterizing the f-vectors of all X of a given type (e.g., of a given homeomorphism type) is hard and mostly unsolved. What has been solved has largely come by way of intriguing combinations of combinatorial, algebraic, and topological methods. This talk will briefly survey the subject; its primary purpose is to provide background for next week's talk.
I will present a truly weird result by Brenti and Welker on the limiting behavior of the f-vectors of simplicial complexes under barycentric subdivision.