Colloquium by José Samper (Max Planck Institut Leipzig): Dual matroid polytopes and the independence complex of a matroid
A shelling order of a simplicial/polytopal complex is an ordering of the top dimensional faces that allows us to understand various properties of the underlying complex (when it exists). Empirically, some shelling orders are better than others in the sense that they are easier to analyze or come equipped with . This is especially notable for complexes that admit many shelling orders, like polytopes and and matroid independence complexes. We propose a strange connection, linking shelling orders of dual matroid polytopes to shelling orders of independence complexes. In particular, we show that several classical theorems about shellability of matroids have geometric interpretations. We use this to address to propose a new strategy for a 1977 conjecture of R. Stanley about face numbers of independence complexes: that the h-vector is a pure O-sequence. The talk is based on joint work with Alex Heaton.