Paul Bressler
University of Haifa, Israel
Intersection Homology for Convex Polytopes.
Abstract: A convex polytope is the convex hull of a finite set of points in a finite dimensional real vector space. The combinatorics of a polytope is embodied by the partially ordered (by inclusion) set of its faces (vertexes, edges, etc.) Certain combinatorial invariants called the g- and the h-numbers were introduced by R.Stanley who stated conjectures concerning their properties motivated by the "dictionary" relating rational convex polytopes (i.e. those whose vertexes have rational coordinates) to projective toric varieties. Up until fairly recently Stanley's conjectures (as well a number of other conjectures concerning the properties of g- and h-numbers of convex polytopes) were proven for rational polytopes using the above mentioned relationship to toric varieties in conjunction with (very difficult) results on the topology of singular varieties. In my talk I will present sone recent developments in this subject culminating in the proof of Stanley's conjectures.