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Metric geometry and the Hirsch conjectureGeometry/Topology
|Start time:||Tue, Oct 2 2012, 3:10PM|
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. In 2010, Santos has disproved the conjecture. So the bound n-d is wrong; but it could be that 2n is the correct guess... 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, and more generally, even for flag homology manifolds. The proof uses a metric criterion by Gromov. If time permits, we will discuss other possible applications of metric geometry.