Speaker: Spencer Backman, HU
Title: Cone valuations, Gram's relation, and flag-angles
Abstract: Interior angle vectors of polytopes are semi-discrete analogues of f-vectors that take into account the interior angles at faces measured by spherical volumes. In this context, Gram's relation takes the place of the Euler-Poincaré relation as the unique linear relation among the entries of the interior angle vectors. Simple normalized cone valuations naturally generalize spherical volumes, and in this talk I will show that Gram's relation is the unique linear relation for angle vectors associated to a cone valuation. Our proof goes by way of establishing a connection with the combinatorics of zonotopes. I will then introduce flag-angle vectors as a counterpart to flag-f-vectors of polytopes, and determine their linear relations by coalgebra methods and a connection to the flag-vectors of lattices of flats. This is joint work with Sebastian Manecke and Raman Sanyal.