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Toric Topology Workshop KAIST 2008
(2008 / vol.10 / no.1)
Lectures on Toric Topology
Victor M Buchstaber
Pages. 1--64   



A convex $n$-dimensional polytope is called simple if there are
exactly $n$ facets meeting at every vertex.
For many decades simple polytopes have been studied in convex geometry
and combinatorics. Recently it has become clear that they play
important role in algebraic and symplectic geometry, in applications
to physics. They are also main objects in toric topology.
There is a commutative associative ring $mathcal{P}$ generated by simple polytopes.
The ring $mathcal{P}$ possesses a natural derivation $d$, which comes from
the boundary operator. We shall describe a ring homomorphism from
the ring $mathcal{P}$ to the ring of polynomials $mathbb{Z}[alpha,t]$
transforming a simple polytope to the face-polynomial and the
operator $d$ to the partial derivative $partial/partial t$.

This result opens way to a relation between theory of polytopes and
differential equations. As it has turned out, certain important
series of polytopes (including some recently discovered) lead to
fundamental non-linear differential equations in partial
derivatives.

We shall discuss constructions of important series of simple
polytopes, and transformations of these series into non-linear
differential equations. Particular examples of the transformations
link Stasheff polytopes (also known as associahedra) to the E.Hopf
equation, and Bott-Taubes polytopes (cyclohedra) to the Burgers
equation.

In the next series of lectures, I will discuss in details many ideas
from this lecture.
end{quote}



Lecture I : Face-polynomials of simple polytopes and applications
Lecutre II : Toric Topology of Stasheff Polytopes
Lecture III : Minkowski sum and simple polytopes
Lecture IV : Moment-angle complexes and applications
Lecture V : Quasitoric manifolds