Victor M. Buchstaber
Polytopes, Hopf algebras and Quasi-symmetric functions.
Tue, 11/09/2010 - 16:01 — SomNambulAuthors: Victor M. Buchstaber, Nickolai Erokhovets
Subjects: Combinatorics
AbstractIn this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
$\mathcal{P}$ generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product $\times $ and a
join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative
associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb
Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative
threegraded ring of polynomials.Ring of Polytopes, Quasi-symmetric functions and Fibonacci numbers.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Victor M. Buchstaber, Nickolai Erokhovets
Subjects: Combinatorics
AbstractIn this paper we study the ring $\mathcal{P}$ of combinatorial convex
polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the
operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all
its $(n-k)$-dimensional faces.Toric Genera.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Victor M. Buchstaber, Taras E. Panov, Nigel Ray
Subjects: Algebraic Topology
AbstractOur aim is to develop topological analogues of an ongoing programme in toric
geometry, which seeks to express arithmetic, elliptic, and related genera of
toric varieties as functions of their fans. In this context, we introduce
methods for computing equivariant genera of omnioriented quasitoric manifolds M
purely in terms of the combinatorial data (P,\Lambda) by which such M are
determined. We develop the theory around the universal example \Phi, which was
introduced independently by Krichever and Loeffler in 1974, albeit from
radically different viewpoints.
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