Poetics, primes, and $\pi_n$.

Dear Elsa,

I'm not today writing so much to say anything particularly mathematical, but to describe something of what it is like to study mathematics, and something of what my own mathematical interests are.

I'm not a historian, but I get the impression that the way maths are studied today is quite different from they way they were studied in, say, 1828, which is in turn different from what you'd have found in 1687, and so on, back to the lovely compilation of surveying tools and book-keeping tricks known as "Euclid's Elements of Geometry". Mathematics also has its fads and fashions, which come and go in various times and places.

Euclid, of course, studied plane geometry, starting from intuitive propositions and building towards elaborate consequences; and that at least has remained as a consistent motif in mathematics. It's an awful lot like building life-sized castles out of toothpicks and paste --- only at some point you don't really see the toothpicks anymore, just the bricks and framing you weave them into. Once you get really good at bricks and framing, you might not notice the toothpicks at all until you start thinking about patterned wallpaper, and realize you need something that isn't a brick, a beam, or a joist.


In fact, a lot of research in mathematics proceeds in a way a lot like saying "something proceeds in a way a lot like something else" --- mathematics is not only about building elaborate things from simple things, but it's also about building elaborate things until they remind us enough of simpler things; again, there is a tendency to build competeing elaborations until they look broadly similar, but with visibly variant elementary bits. To put it more poetically, mathematics is the precision study of analogy, largely motivated by appealing analogies.

Now, as any poet knows, metaphor and simile are litterary, not litteral, and that's part of what makes them beautiful. Chersterton's slipper hunt is only as fun as it is because we know slippers don't really run away on their own, as much as we may imagine so trying to find them the rest of the time. Similarly, as other mathematicians have written (Baez and Connes come to mind), the more provocative analogies pursued by mathematicians are those that don't reduce to strict equivalence.

Just to illustrate, I'd like to describe a (very mathematical) analogy that doesn't approach any kind of precision, but still keeps me entertained. To do that, I'll have to tell you two stories, and, well ... we'll see how it goes.

There are several popular ways to name a natural number. One dull way is just to give your friend a bag with that many pepper imps in it; this method has its obvious limitations. Another way is to describe the number as a sum of fibonacci numbers, for example

$ 82 = 1 + 5 + 21 + 55 $

which gets every number; you can even insist that you only use each fibonacci number once (or not at all), and never use consecutive fibonacci numbers, and then there is still exactly one way to describe any natural number. So a number can effectively be named as strung together from bits chosen from $'fs'$, $'s'$; $'.'$ and $'f.'$: if you write $'.'$ or $'f.'$ you stop; then the number you know as $82$ looks like $'fsfssfssf.'$. Here, the last $f$ or $s$ says whether the fibonacci number $1$ is needed; the one before says whether $2$ is needed, then $3$, and so on. It's even very simple to figure out which of the fibonaccis you need: just find the biggest fibonacci that isn't bigger than your number (in the case of describing 82, it's 55), and work out what you need to describe the difference${}^1$ $(82-55)=27$.

Another way to describe a natural number is as a product of irreducible numbers --- commonly known as primes. In this system, we have (for instance) $82=2\times 41$. This system has properties very useful for lots of algebraic nonsense (group theory, cryptography, ... ) but it's also got a fair ammount of difficulty to it; for instance, one needs to know what the primes are. Another way the primes versus the fibonaccis are different is that the primes are a great deal thicker than the fibonaccis, because multiplying numbers makes them get big much quicker than does adding them. They do both thin out as they get larger, but the primes are definitely weirder. (And this does help for the cryptographic application described above). Notably, it takes much longer to factor a generic natural number into its primes than it does to write it as a sum of nonrepeating nonconsecutive fibonacci numbers.

That's one of my two stories. The other one is about... well, I'd say it was about spaces, but that's both too broad a word and yet also doesn't quite suggest how weird "spaces" can be. A better word would probably be "geometric figures", where we don't mind how wobbly our drawing is --- and we also pretend we can draw arbitrary-dimension objects.

Perhaps a more descriptive metaphor for my kind of "spaces" would be "sewing projects". One can, for instance, treat both armchairs and tire inner-tubes as elaborate sewing projects, involving three-dimensional things (chair stuffing, or compressed air for inner tires, although sometimes...) and two-dimensional things (the sheets of cloth/leather you need) sewn together along one-dimensional things (seams, held together by thread) which come to a terminus at zero-dimensional things (knots in the thread, usually, but here we may be stretching the analogy a bit much...). So you're already used to four steps (or gradings) in the ladder of dimensions for our figure-sewing projects.

Now, as you know chairs and inner-tubes are rather different shapes. You will recognize this when I tell you that chair-shapes incorporating inner-tube shapes fall into two distinct categories: those for people without practical use of their legs, and those made out of porcelain.

But on a more fundamental level, they have different topologies. Now, in the poetic and vague mathematical I'm setting up, I've only got so far as describing the analogue of numbers and fibonacci numbers. OK, I now admit that the fibonacci numbers were kind of silly... powers of $10$ will do nearly as well --- but they do rather emphasize that while there may be many ways to write a number as a sum of fibonaccis (we can, of course, consider $55+55=110$), but that some are better than others ($55+55=55+34+21=89+21$) and the best way have special properties. In the numbers and fibonacci case, this means we don't even need consecutive fibonacci numbers; in the sewing project case, it means we can always suppose that a 3-dimensional thing is sewn (stuffed or glued) along its boundary to a 2-dimensional thing; and so on up or down the ladder.

This very nice and convenient way of describing sewing-project figures in fact corresponds to a handy algebraic gadget --- named $h_n(X)$ for natural numbers $n$ and sewing project $X$ --- which helps classify our spaces, at least so far as putting them in the right chapters of a dictionary (oh my, how mixed up our metaphors are getting!); and the possible chapters of our dictionary are called the homology coalgebras. (This in turn is related to one of Dr. Thursday's FSA board games, which maybe I'll write about another time.) There is, however, a different way of describing a sewing project, very different indeed, which is much trickier to get any handle on from a visual point of view, but which in many ways is more fundamental, and are even called so.

Given any sewing project figure (suppose it's a chair), we can make a new one by just sewing in a sheet along its edge --- and the edge is just a circle. Being mathematicians (I hope youu don't mind being called one?), we're interested in what two ways of sewing-on a sheet look different. For that context I'm most curious about, two such ways of sewing a sheet to a chair are "the same" (or "equivalent") if you can
\begin{itemize}
\item connect their seams by a curve on the chair and then
\item glue a patch onto the chair so that it's edge follows one seam, the connecting curve, the other seam, and then backwards along the connecting curve again.
\end{itemize}
I won't get into details here, but we've just described the basic ingredients for the fundamental functors $\pi_0$ and $\pi_1$. $\pi_0$ considers the same any two spots you can join with a curve, and $\pi_1$ considers the same any closed curves you can fill-in with a patch --- although there are a few more rules and wrinkles --- and the story also continues up the ladder, so for each sewn figure $X$ and each natural number $n$ we have an object $\pi_n(X)$.

The homologies $h_n(X)$ are like an abbreviation of how our project is sewn from sheets of different sizes; the fundamental functors $\pi_n(X)$ describe how the same project can be combed out${}^2$ with "simple" projects as typical strands. Of course, "simple" here means something quite complicated --- except as seen by $\pi_n$! The "simple" projects are usually called $K(n,G)$ where $G$ is a sufficiently nice, easily-understood thing --- never mind that now --- and $n$ is a natural number. The defining characteristic of a $K(n,G)$ (there are many, but they're similar enough) is that $\pi_n(K(n,G)) = G$ and otherwise $\pi_m(K(n,G))=\{1\}$ whenever $m\neq n$.

Anyway, in the poetic analogy I'm imagining of late, combing-out is very like factoring numbers, and the sort of threads you get, the $K(G,n)$s, are very like prime numbers in this regard --- not only in that there's not much combing left to be done on them, but also that it takes some work to understand them and pick them out from among all the other objects that might interest us. (I wonder if we ought to look for a Riemann hypothesis of homotopy types?)

I've been writing this for five days, now, and I seem to have lost something of my rhythm, and I don't know how to wrap up... but that's about all I wanted to say. Well, I hope you've enjoyed it, anyways.

an awkward fellow

${}^1$ this sort of "greedy" algorithm, to find the best simple fit you can and then make it better, is another common trick of the precision study of analogy.

${}^2$ this is somewhat like combing out a jaguar pelt. There will probably be some twists/tangles to undo, and it might not look spotted afterwards!