# Manchester Geometry Seminar 2009/2010

**Thursday 25 February 2010. ** *The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm*
##
Polytopes, Hopf Algebras, and Quasi-Symmetric Functions

Victor Buchstaber (Steklov Mathematical Institute and University of Manchester)

`buchstab@mi.ras.ru`

Differential ring of convex polytopes was introduced by the speaker some time ago. As we have recently discovered, this ring possesses additional rich algebraic structure, in particular a Hopf comodule structure.

There are several canonical maps sending a convex polytope to a sum of quasi-symmetric monomials, whose coefficients completely define the flag vector of the polytope. They transform the algebraic structures on the ring of polytopes to the corresponding structures on the ring of quasi-symmetric functions.

This opens ways to applying technique and results of Hopf algebras and quasi-symmetric functions to studying combinatorial polytopes. We shall describe results about flag vectors of convex polytopes obtained by this new approach.

The talk is based on the recent work with
N. Erokhovets, published partially in:

V. M. Buchstaber, N. Yu. Erokhovets, *Ring of polytopes, quasi-symmetric functions and Fibonacci numbers*,
`arXiv: 1002.0810 v1 [math CO] 3 Feb 2010`

.