Combinatorics Seminar 2017

Speaker: Anton Ayzenberg, Higher School of Economics

Date: Thu, Nov 30, 2017

Time: 11:00 - 12:00

Venue: Large Meeting Room

Title: Algebras of multi-polytopes: the geometry behind combinatorics of triangulated manifold


Given a triangulated $(n-1)$-sphere, one can count the number of its vertices, the number of its edges, etc., thus obtain the $f$-vector. The long-standing problem in combinatorics is the characterization of all possible $f$-vectors of spheres. It is common to use $h$-vectors instead of $f$-vectors. The $h$-vectors carry the same information but for spheres they are better behaved: they are symmetric and nonnegative. This result relies in essential way on a fact that the Stanley-Reisner algebra of a triangulated sphere is Gorenstein. Many attempts were made to generalize this theory to arbitrary triangulated manifolds. For triangulated manifolds it is common to use $h''$-numbers, which are linear combinations of $h$-numbers and Betti numbers of a manifold. Novik and Swartz made a breakthrough in 2009, by showing that $h''$-numbers are nonnegative and symmetric. In their proof they constructed certain Gorenstein quotient of the Stanley-Reisner algebra. In my talk I will show the interpretation of $h''$-numbers in terms of convex geometry. We will consider multi-polytopes: non-convex generalizations of polytopes. The $h''$-numbers can be extracted from the volume polynomial of a multi-polytope and the ring of differential operators. Relations in Novik-Swartz algebra can be seen as Minkowski relations for a multi-polytope.

The talk is based on a joint work with Prof. Mikiya Masuda from Osaka City University.