SPEAKER:  Spencer Backman  (Hebrew University of Jerusalem)

TITLE:  Cone valuations, Gram's relatin, 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.