Bruno Benedetti: Some recent progress on the Hirsch conjecture

A d-dimensional simplicial complex with n vertices is called "Hirsch" if its dual graph has diameter smaller than n-d. The Hirsch conjecture (1957) asked whether the boundary of every (d+1)-polytope is Hirsch. (The question came from optimization, as a worst-case scenario from the simplex algorithm.) In 2010, Santos has disproved the conjecture. 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 correct? We really don't know much: At the moment we don't even have a *polynomial* upper bound in n and d.

We will present some recent progress (joint with Karim Adiprasito): The conjecture holds true for flag polytopes. (With the original bound, n-d-1). More generally, flag spheres, flag manifolds and even flag complexes with strongly connected links, are all Hirsch. The proof uses a metric criterion by Gromov.