Raman Sanyal: Gram’s relation, cone valuations, and combinatorics

Tid: On 2018-02-14 kl 10.15 - 11.15

Föreläsare: Raman Sanyal, Goethe-Universität Frankfurt

Plats: Room 3418, Lindstedtsvägen 25. Department of Mathematics, KTH


Euler’s Polyhedron Formula and it’s generalization, the Euler-Poincare formula, is a cornerstone
of the combinatorial theory of polytopes. It states that the number of faces of various dimensions
of a convex polytope satisfy a linear relation and it is the only linear relation (up to scaling).
Gram’s relation generalizes the fact that the sum of (interior) angles at the vertices of a convex
$n$-gon is $(n-2)\pi$. In dimensions $3$ and up, it is necessary to consider angles at all faces. This
gives rise to the interior angle vector of a convex polytope and Gram’s relation is the unique linear
relation (up to scaling) among its entries. In this talk, we will consider generalizations of “angles” in
the form of cone valuations. It turns out that the associated generalized angle vectors still satisfy
Gram’s relation and that it is the only linear relation, independent of the notion of “angle”! To prove
such a result, we rely on a very powerful connection to the combinatorics of zonotopes and
hyperplane arrangements. If time permits, we will ponder the notion of flag-angle vectors, a
semi-discrete counterpart to flag-vectors of polytopes, and the linear relations satisfied by flag-angle
vectors. This is joint work with Spencer Backman and Sebastian Manecke.