Differential, Dude

The most common criticism I receive about my work is that it can't possibly have anything to do with physics, because there are no differential equations. So it is with great delight I discover that V. Buchstaber at Manchester is working on turning polytope combinatorics into interesting partial differential equations.

First, consider only simple polytopes. That is, ones in $d$ dimensions with $d$ faces meeting at a vertex. For example, the three dimensional Stasheff associahedron has 3 faces (pentagons or squares) meeting at each vertex. Now group equivalent polytopes into classes (a common trick) and then make an algebra from combinations of these classes. The zero is the empty polytope and the unit is the single point. There is an operator $D$ that sends a $d$ dimensional polytope to a $(d - 1)$ dimensional one. For example, on the simplex $K_{n}$ it acts as

$D K_{n} = (n + 1) K_{n - 1}$

sending a $4$-simplex to the 5 tetrahedra on its boundary. Let $f_{k, n-k}$ denote the number of $k$ dimensional faces of an $n$ dimensional polytope. Then for any such polytope $P$ there is a homogeneous polynomial in $a$ and $t$ given by

$F(P) = a^{n} + f_{n-1,1} a^{n-1} t + \cdots + f_{0,n} t^{n}$

Buchstaber shows that the map $F$ satisfies

$F(DP) = \frac{\partial}{\partial t} F(P)$

An interesting sequence $P_{n}$ of polytopes turns out to be the sequence of associahedra. In this case, by letting

$U(a,t,x) = \sum_{n} F(P_{n}) x^{n+2}$

it turns out that $U(t,x)$ must be the solution of the Hopf equation

$\frac{\partial}{\partial t} U(t,x) = U(t,x) \frac{\partial}{\partial x} U(t,x)$

with $U(0,x) = x^{2} (1 - ax)^{-1}$. This is related to the important KdV equation from soliton theory.