Some recent progress on the Hirsch conjecture

A polytope is the convex hull of finitely many points in R^d. Given two arbitrary vertices of a d-dimensional polytope with n facets, how far away can they be? That is, how many edges do we have to walk on, to go from one to the other? The question comes from optimization, as a worst-case scenario from the simplex algorithm. Let us say that a d-dimensional simplicial complex with n vertices is "Hirsch" if its dual graph has diameter smaller than n-d. The Hirsch conjecture (1957) guessed that the boundary of every (d+1)-polytope is Hirsch. The conjecture has recently been disproved by Paco Santos (Annals of Math., 2012). So the bound n-d is wrong; but it could be that 2n is the correct guess; or maybe 2n is also wrong, but dn is the correct one... We really don't know much: At the moment we cannot even prove a *polynomial* upper bound in n and d. We will present some recent progress (joint work with Karim Adiprasito): The Hirsch conjecture holds true for flag polytopes, and more generally, even for flag cohen-Macaulay complexes. In particular, the barycentric subdivision of every triangulated manifold is Hirsch. The proof uses a metric geometry criterion by Gromov, but is otherwise elementary; I will start from defining what a simplicial complex is.