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\begin{document}

\title{Polytopes, Hopf algebras\\ and quasisymmetric functions.}
\author{Victor M. Buchstaber}
\address{Steklov Mathematical Institute, Russian Academy of
Sciences}
\email{buchstab@mi.ras.ru}
\address{School of Mathematics University of Manchester, UK}
\email{Victor.Buchstaber@manchester.ac.uk}

\maketitle
\begin{center}
\textbf{Abstract}
\end{center}



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. % over a certain (newly constructed) Hopf algebra.


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:

\noindent V.~M.~Buchstaber, N.~Yu.~Erokhovets,  \emph{Ring of
polytopes, quasisymmetric functions  and  Fibonacci numbers.},
arXiv: 1002.0810 v1 [math CO] 3 Feb 2010.



\end{document}
