### 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.

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.

## 6 Comments:

Your comment on Woit's blog (and on mine) were welcome. They could level the same charge at Witten that his work has nothing to do with physics. They could also criticise Paul Steinhardt and others for attacking inflation, but think it easier to pick on the girl. The geometries in your posts could hsve much to do with physics, and could also lead to insights like Gell-Mann's eightfold way. Keep up the good work!

Thanks for your support, Louise. Yes, I am frequently criticised and only very, very rarely commended, which seems to be a common condition for my gender. I would appreciate the criticism more if it came from people who actually had some clue what I was talking about, but strangely enough, such people tend to be encouraging. I have thus learned over the decades, despite my thick skull, not to pay much attention to what anybody thinks.

P.S. The Riess talk was also interesting. His method for direct fixing of Hubble H0 in the 70s (against the lower Sandage result favoured by Wiltshire) appears to be quite robust. They have used fixed Cepheid types (and the same telescope) to calibrate the distance scale and eliminate some systematics.

Hi Kea,

I have not read PW for a long time, but will, to learn about your referenced comment.

The Buchstaber paper is interesting.

Note that he also had training at the Russian Academy of Science as did Kozloz and Ilinski; then to England as Ilinski.

05 12 08

Kea:

Thanks very much for the link to the PDF file. I have been on the periphery of understanding SOME of the stuff you talk about for a while, but without a true understanding of the mathematics per se. This paper you directed us to acts as a Rosetta stone connecting the diff eqs to categorical type thinking and I appreciate it a great deal. Now, my understanding is slightly increased.

I don't know why someone would say that your work ain't physics because there are no diff eqs! How unreasonable. It is all about perspective here. If you are working on new fangled mathematics that represent the physical universe, then I believe that is physics, just coming from another perspective.

So thanks for the link. I hope that by reading this stuff, my fetus will become smart!

Mahndisa, good to hear from you! I guess it can't be long til the kid comes? How exciting!

I'm afraid the mathematics will

alwaysbe difficult: there is simply too much involved for one person to take in, even for people much smarter than myself. Actually, soliton theory was one of the things that got me into quantum algebra and category theory in the first place. About 15 years ago I took some courses on solitons, quantum groups and the like, which was all about solving differential equations!Post a Comment

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