A simplicial polytope P is the convex hull of finitely many points in R^d such that all the faces of P are simplices. To each such polytope, we can associate its f-vector, which is the integer vector recording the number of faces of each dimension. McMullen conjectured that the f-vectors of simplicial polytopes are completely characterized by three strikingly simple conditions. The sufficiency of these conditions was proved by Billera and Lee, and the necessity was proved by Stanley using Hodge theory.
In this talk, we will review the construction of toric varieties from polytopes and intersection cohomology. We will then give a brief account of Stanley's proof of the necessity part of the g-conjecture.