The Unapologetic Mathematician

Zeno, who’s Halfway There, has a great little story about how his students react to mixed fractions.

I, on the other hand, got such gems as “f*** this I’m withdrawing anyway” on my recent exam. Is it good that my students know enough not to even try to butter me up, or am I missing out on the love of undergraduates everywhere?

Well I thought I had the perfect one this week, but then I watched its video and remembered why Bauhaus is held up as the pinnacle of all the over-indulgent post-punk out there. So that’s not happening.

But while I was watching, I also got to thinking: Nouvelle Vague must have also recognized the depths that lay to be plumbed here. And sure enough, they had. It’s not as much of the bossa nova sound, but it’s still a worthy reworking of the original. From their 2006 album Bande à Part: Bela Lugosi’s Dead.
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I think I’m done with categories qua categories for now, and am ready to move on to another subject for a little while. But before I do, I’m going to take a little break and finally get down to this restructuring of the subjects listed on the right. It’ll also give me some time to catch up on some stuff in the real world that needs doing.

The RSS feed will probably be going nuts with updates to posts as I crawl through the archives and relabel things. My apologies in advance.

[UPDATE]: I’ve finished the group theory archives. Unfortunately, the “category” bar on the right seems to not indent nested categories in this theme. Sort of annoying…

[UPDATE]: There seems to be a bug with subcategories on WordPress today, and I’m not the only one having it. So I’ll have to hold off on refining the rings and categories topics until later. In the meantime, I’ve reworked the sidebars a bit, including a search field!

This week, I think I’ll continue the theme of great videos with some of the best breakdancing out there. DJ Format (featuring Jurassic 5) with We Know Something.
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I ran across an odd little page today: an ad selling a restoration of Fermat’s proof of FLT. Surely it’s either a hoax, or not what the ad claims it to be. However, it would be nice to see exactly what sort of snake oil they’re selling.

There’s a new paper on SSRN suggesting a dress code for law professors. And, frankly, I’d be for a similar norm (social, not compulsory) in the rest of academics. For the reasons, go read the paper.

Thankfully, the author even includes climate modifications for those (like me) who can’t yet stand a jacket and tie in a gulf coast September. One thing that was nice about the visit to D.C. was being able to wear such again without dropping dead of heatstroke.

The one thing about the paper that makes me wonder is the abstract, which asserts that “Law professors dress scruffily”. In my experience, though, it’s not really the case.

Okay, so I’ll pile on with the interactive fiction chatter. I really should, since I’ve been playing IF games since I was a wee lad.

First someone pointed out that a long calculation is like a computer game where you have to save and keep backtracking to your saved states. Isabel at God Plays Dice then drew the more specific connection to interactive fiction. Then Mark at Inductio Ex Machina contributed this sample transcript of such a “game”.

I’d like to point out an unintended analogy here. It’s pretty well accepted within the IF community that any puzzle should be solvable at a first pass. That is, if you’ve done everything right you’ll have all you need to solve a given puzzle without guessing, failing, and backtracking. In fact, it’s the height of bad writing to include a puzzle that requires you to attempt it and fail to gain information needed to pass.

I think that the same holds true in mathematics. If you find that you must do a hard calculation with attendant backtracking, you’re asking the wrong question. When properly viewed, the solution to any problem should be inherent in the problem itself. Of course, it might be more convenient in context to bash your head against a wall than to look for the hidden doorway, but it’s really not the best way to go about things in the long run. I come back to my favorite passage from Grothendieck’s Récoltes et Semailles.

Prenons par exemple la tâche de démontrer un théorème qui reste hypothétique (à quoi, pour certains, semblerait se réduire le travail mathématique). Je vois deux approches extrêmes pour s’y prendre. L’une est celle du marteau et du burin, quand le problème posé est vu comme une grosse noix, dure et lisse, dont il s’agit d’atteindre l’intérieur, la chair nourricière protégée par la coque. Le principe est simple: on pose le tranchant du burin contre la coque, et on tape fort. Au besoin, on recommence en plusieurs endroits différents, jusqu’à ce que la coque se casse — et on est content. Cette approche est surtout tentante quand la coque présente des aspérités ou protubérances, par où “la prendre”. Dans certains cas, de tels “bouts” par où prendre la noix sautent aux yeux, dans d’autres cas, il faut la retourner attentivement dans tous les sens, la prospecter avec soin, avant de trouver un point d’attaque. Le cas le plus difficile est celui où la coque est d’une rotondité et d’une dureté parfaite et uniforme. On a beau taper fort, le tranchant du burin patine et égratigne à peine la surface — on finit par se lasser à la tâche. Parfois quand même on finit par y arriver, à force de muscle et d’endurance.

Je pourrais illustrer la deuxième approche, en gardant l’image de la noix qu’il s’agit d’ouvrir. La première parabole qui m’est venue à l’esprit tantôt, c’est qu’on plonge la noix dans un liquide émollient, de l’eau simplement pourquoi pas, de temps en temps on frotte pour qu’elle pénètre mieux, pour le reste on laisse faire le temps. La coque s’assouplit au fil des semaines et des mois — quand le temps est mûr, une pression de la main suffit, la coque s’ouvre comme celle d’un avocat mûr à point ! Ou encore, on laisse mûrir la noix sous le soleil et sous la pluie et peut-être aussi sous les gelées de l’hiver. Quand le temps est mûr c’est une pousse délicate sortie de la substantifique chair qui aura percé la coque, comme en se jouant — ou pour mieux dire, la coque se sera ouverte d’elle-même, pour lui laisser passage.

L’image qui m’était venue il y a quelques semaines était différente encore, la chose inconnue qu’il s’agit de connaître m’apparaissait comme quelque étendue de terre ou de marnes compactes, réticente à se laisser pénétrer. On peut s’y mettre avec des pioches ou des barres à mine ou même des marteaux-piqueurs: c’est la première approche, celle du “burin” (avec ou sans marteau). L’autre est celle de la mer. La mer s’avance insensiblement et sans bruit, rien ne semble se casser rien ne bouge l’eau est si loin on l’entend à peine… Pourtant elle finit par entourer la substance rétive, celle-ci peu à peu devient une presqu’île, puis une île, puis un îlot, qui finit par être submergé à son tour, comme s’il s’était finalement dissous dans l’océan s’étendant à perte de vue…

Le lecteur qui serait tant soit peu familier avec certains de mes travaux n’aura aucune difficulté à reconnaître lequel de ces deux modes d’approche est “le mien” — et j’ai eu occasion déjà dans la première partie de Récoltes et Semailles de m’expliquer à ce sujet, dans un contexte quelque peu différent. C’est “l’approche de la mer”, par submersion, absorption, dissolution — celle où, quand on n’est très attentif, rien ne semble se passer à aucun moment: chaque chose à chaque moment est si évidente, et surtout, si naturelle, qu’on se ferait presque scrupule souvent de la noter noir sur blanc, de peur d’avoir l’air de combiner, au lieu de taper sur un burin comme tout le monde… C’est pourtant là l’approche que je pratique d’instinct depuis mon jeune âge, sans avoir vraiment eu à l’apprendre jamais.

In case you haven’t yet passed your French language qualifier, I’ll give a rough translation.

Take, for example, the task of proving a theorem. I see two extreme approaches one could take. The first is that of hammer and chisel, wherein the problem posed is seen as a large nut, hard and smooth, which contains a nourishing meat protected by the shell. The principle is simple: one puts the edge of the chisel against the shell and hits it hard. If necessary, one tries again in many different places, until the shell cracks — and one is happy. This approach is especially appealing when the shell shows a rough or bumpy patch where it can be grasped. In some cases, such places to grab the nut jump to the eye. In other cases, one must use all one’s senses and search carefully before finding a point of attack. The most difficult case is that where the shell is perfectly round and evenly firm. When hit strongly, the edge of the chisel just scratches the surface — one ends up merely tired. Sometimes the nut will finally crack through mere strength and stamina.

I can illustrate the second approach with the same metaphor of a nut to be opened. The first explanation that comes to mind is to immerse the nut in some softening liquid — water, for instance — and to rub it from time to time to allow the water to penetrate better, but otherwise to leave it alone. Over weeks and months, the shell softens — when the time is right, a flick of the wrist is sufficient, and the shell opens to it like a ripe avocado! Or again, one can leave the nut out in the sun and the rain and even through the icy winter. When the time is right, it is a delicate touch that breaks the shell — or to say it better, the shell will open itself to let one through.

The pictur that came to me recently was again different. The unknown thing one is trying to undertand seems to me like a stretch of land or a hard patch of earth, hard to dig into. One might go at it with picks or mining tools, or even with jackhammers: this is the first approach, that of “chisels” (with or without hammer). The other is that of the sea. The sea advances imperceptibly and noiselessly. Nothing seems to break, nothing moves… Yet eventually it surrounds the land. It slowly becomes a peninsula, then an island, then an islet, and finally it is submerged completely, dissolved into the ocean which stretches as far as the eye can see…

The reader who is familiar with some of my work will have no difficuly determining which of these two approaches is “mine” — and I have had occasion already in the first part of these “Reapings and Sowings” to explain myself on this subject, in a slightly different context. It is “the method of the sea”, by submersion, absorption, dissolution — that where, if one does not pay close attention, nothing seems to happen at any given moment: everything is at each moment so evident and so natural that one feels nervous to write it down in black and white for fear of being others’ disapproval, rather than banging away at a chisel like everyone else… Yet this is the approach that I instinctively took since I was young, never having really noticed it.

As for the title of this post, a feelie is a physical object — often some document — that was packaged with a game and containing information crucial to some puzzle you’d need to solve. That is, if you didn’t buy the game and get the feelie, you couldn’t get past a certain point. It provided a crude level of copy-protection back in the good old days, under the pretense of extending the game experience (more common in non-IF games was asking the user to type in some specified word from the documentation). Thus, a feelie was all too often a hack — a puzzle relying on them was awkward and inelegant, pulling you out of the experience of the game rather than immersing you in it as was hoped.

Blackboards full of equations serve the same obscuring purpose. True understanding never lies in a calculation. The chalk on the board should not be a map, but a lens, and the mathematics is not in the equations, but behind them.

Sorry about going AWOL yesterday, but I got bogged down in writing another exam.

Okay, so we’ve set out chain homotopies as our 2-morphisms in the category of chain complexes in an abelian category . We also know that each of these 2-morphisms is an isomorphism, so decategorifying amounts to saying that two chain maps are “the same” if they are chain-homotopic.

Many interesting properties of chain maps are invariant under chain homotopies, which means that they descend to properties of this decategorified version. Alternately, some properties are defined by 2-functors, which means that if we apply a chain homotopy we change our answer by a 2-morphism in the target 2-category, which must itself be an isomorphism. I like to call these “homotopy covariants”, rather than “invariants”. Anyway, then the decategorification of this property is an invariant, and what I said before applies.

The big one of these properties we’re going to be interested in is the induced map on homology. Let’s consider chain complexes and , chain maps and from to , and let’s say there’s a chain homotopy . The chain maps induce maps and . I assert that .

To see this, first notice that passing to the induced map is linear. That is, . So all we really need to show is that a null-homotopic map induces the zero map on homology. But if makes null-homotopic, then . When we restrict to the kernel of , this just becomes , which clearly lands in the image of , which is zero in , as we wanted to show.

Now if we have chain maps and along with chain homotopies and , we say that and are “homotopy equivalent”. Then the induced maps on homology and are inverses of each other, and so the homologies of and are isomorphic.

This passage from covariance to invariance is the basis for why Khovanov homology works. We start with a 2-category of tangles (which I’ll eventually explain fore thoroughly). Then we pick a ring and consider the 2-category of chain complexes over the abelian category of -modules. We construct a 2-functor that picks a chain complex for each number of free ends, a chain map for each tangle, and a chain homotopy for each ambient isotopy of tangles. Then two isotopic tangles are assigned homotopic chain maps — the chain map is a “tangle covariant”. When we pass to homology, we get tangle invariants, which turn out to be related to well-known knot invariants.

We’ve defined chain complexes in an abelian category, and chain maps between them, to form an -category . Today, we define chain homotopies between chain maps, which gives us a 2-category.

First, we say that a chain map given by is “null-homotopic” if we have arrows such that . Here’s the picture:

In particular, the zero chain map with for all is null-homotopic — just pick .

Now we say that chain maps and are homotopic if is null-homotopic. That is, . We call the collection a chain homotopy from to . Then a chain map is null-homotopic if and only if it is homotopic to the zero chain map. We can easily check that this is an equivalence relation. Any chain map is homotopic to itself because and the zero chain map is null-homotopic. If and are homotopic by a chain homotopy , then is a chain homotopy from to . Finally, if is null-homotopic by and is null-homotopic by , then is null-homotopic by .

Another way to look at this is to note that we have an abelian group of chain maps from to , and the null-homotopic maps form a subgroup. Then two chain maps are homotopic if and only if they differ by a null-homotopic chain map, which leads us to consider the quotient of by this subgroup. We will be interested in properties of chain maps which are invariant under chain homotopies — properties that only depend on this quotient group.

In the language of category theory, the homotopies are 2-morphisms. Given 1-morphisms (chain maps) , , and from to , and homotopies and , we compose them by simply adding the corresponding components to get .

On the other hand, if we have 1-morphisms and from to , 1-morphisms and from to , and 2-morphisms and , then we can “horizontally” compose these chain homotopies to get with components . Indeed, we calculate

We could also have used and done a similar calculation. In fact, it turns out that and are themselves homotopic in a sense, and so we consider them to be equivalent. If we pay attention to this homotopy between homotopies, we get a structure analogous to the tensorator. I’ll leave you to verify the exchange identity on your own, which will establish the 2-categorical structure of .

One thing about this structure that’s important to note is that every 2-morphism is an isomorphism. That is, if two chain maps are homotopic, they are isomorphic as 1-morphisms. Thus if we decategorify this structure by replacing 1-morphisms by isomorphism classes of 1-morphisms, we are just passing from chain maps to homotopy classes of chain maps. In other words, we pass from the abelian group of chain maps to its quotient by the null-homotopic subgroup.

As promised, something lighter.

Okay, a couple weeks ago I defined a chain complex to be a sequence with the property that . The maps are called the “differentials” of the sequence. As usual, these are the objects of a category, and we now need to define the morphisms.

Consider chain complexes and . We will write the differentials on as and those on as . A chain map is a collection of arrows that commute with the differentials. That is, . That these form the morphisms of an -category should be clear.

Given two chain complexes with zero differentials — like those arising as homologies — any collection of maps will constitute a chain map. These trivial complexes form a full -subcategory of the category of all chain complexes.

We already know how the operation of “taking homology” acts on a chain complex. It turns out to have a nice action on chain maps as well. Let’s write for the kernel of and for the image of , and similarly for . Now if we take a member (in the sense of our diagram chasing rules) so that , then clearly . That is, if we restrict to , it factors through . Similarly, if there is a with , then , and thus the restriction of to factors through .

So we can restrict to get an arrow which sends the whole subobject into the subobject . Thus we can pass to the homology objects to get arrows . That is, we have a chain map from to . Further, it’s straightforward to show that this construction is -functorial — it preserves addition and composition of chain maps, along with zero maps and identity maps.