More topological reflections of arithmetic

For this part of the story I want a really-pretty diagram that even the mathjax still doesn't understand yet how to draw, so I've prepared a dead picture of it.

The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.

The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!

There are lots more nifty diagrams involved, probably the prettiest being

--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...

Anyway, if anyone wants more, just ask me!

Apocrypha Topologica II

Dear Mathematicelli,

If I may proceed,

The Cretan king Minos seems to have hit upon the psychological trick of bewildering his prisoners into imagining a topological obstruction where in fact the obstacle was only metric: that of the Labyrinth (with intimidating monster to keep you distracted). The solution that let Theseus escape the Labyrinth has found renewed popularity of late, and so is worth elaborating. Ariadne's reasoning might conceivably have run thus:

• If one enters the labyrinth and comes to its center, then one has got (and can get) from there to here

• Walking the same way backwards, one can get there from here

• What is wanted is some means to remember the path one took in getting here, and then to reverse it

• Since I don't know how long a path one might need to follow, the keeper of the memory had better be long!

In other words, we keep more information than just the fact that two points are connected by a path: we remember the whole path between them. Some recenter mathematicians more inclined to vandalism have suggested painting the walls of the labyrinth to remind yourself of where you have been --- which is sufficient data to escape, if you paratroop into the maze under cover of darkness; but the solution proposed by Ariadne and adopted by Theseus makes it easier to tidy-up afterwards: unroll a string as you walk along, and then follow it in reverse, winding the string again!

A point worth noting, which may have escaped Ariadne and Theseus in their flight, is that the re-winding and the follow-backwards phases of the solution can be performed in either order if only the string itself is slippery enough --- this could have been remarkably handy if the labyrinth had been under water, and had they wanted to get fish out of it without the fish seeing them and becoming suspicious. It also points out something special in the total path followed by Theseus through the Labyrinth: it is contractible!

Dungeons we may pass over, as well as the castle and siege warfare. Chain mail is about as old. But Somewhen between the Visigoths and Polyphony was discovered knitting. These, like most mechanical inventions, rely on metric phenomena to make their topology useful, but it is unquestionably their underlying topology that is used. (As a side-note, there's a lovely film-reference back to Ariadne in The Name of the Rose, where Adso returns his knit sweater to a trivial topology in order to escape a non-planar labyrinth! I'm curious how the Vandal painters would fare here!) If you'll forgive a jump-forward, the topologist Poincaré seems to have observed a woman knitting, and independently invented purling on the spot. I don't know what the full topological significance of that is, but the ubiquity of What Is certainly makes itself plain to those who can see it; for which give thanks to God, I think. About such other oddities as the Borromean rings we have remarked elsewhere.

I have no idea what's coming next; but this isn't a bad lot. We've more-or-less covered ±1800. Prof. Cauchemar

More Pythagoras than Pythagoras

So, a nifty theorem: that in a tetrahedron of which the three angles at one vertex are right angles, the square of the opposite face is the sum of the squares on the three adjacent faces. By homogeneity, it follows that the square of any plane area in 3-space is the sum of the squared areas of its three orthogonal shadows on any three orthogonal planes. Here, by homogeneity I mean that orthogonal shadows of parallel plane figures are proportional to eachother as those plane figures are.

A sketch of the hypothesis, as stated:

The "opposite" face is left implicit; the adjacent faces coloured blue, red, and grey.

There are, of course, several approaches to this theorem.

One might calculate relations among the sides and areas, and proceed algebraically to conclude that the two quantities described have equivalent expressions (e.g., in terms of the three adjecent edge lengths).

Alternatively, one might take as prior what we have stated as corollary, and argue (by suitable means) that the sum-of-squared-shadows is independent of which three orthogonal planes are chosen for catching the shadows; then, since the equation obviously holds when our plane figure lies in one of the three planes --- for then it is its own shadow on its own plane, and its other two shadows have no area, and the full proposition follows.

I rather like circles, lately, so I'll do something between the two: a special plane shape (indeed a circular disc) has three orthogonal shadows on three orthogonal planes, and their squares sum to the square on the circle; one returns to general plane figures by homogeneity again.

The niftyness of this arrangement begins with the observation that the orthogonal shadow of a circular disc is an ellipse; and moreover that the major axis of this ellipse equals the diameter of the circle. Indeed, the shadow doesn't depend on where the disc lies, only on its relative attitude. Supposing the disc's center were on the shadow plane, then the shadow and disc meet in a line, precisely this major axis. Since the area of an ellipse is universally proportional to the product of its principal axes, this reduces the problem to arguing that the squares on the minor axes of the three elliptical shadows sum to the square of the circle's diameter.

To get there, we must be sneaky: consider a line segment perpendicular to our circular disc, and equal to a diameter. With some thought, one can see (this means "prove", but it's not too hard) that the minor axis and the shadow of this perpendicular lie on the same line; but they are shadows of equal and perpendicular line segments, so their squares must sum to the square on the diameter --- once in each ellipse.
At the same time, the three orthogonal shadows of the perpendicular segment squared sum to twice the square on the diameter (this is a fun exercise) adding up six squared line shaddows in three ellipses should give three times the squared diameter (since each ellipse gives one), so the three squared minor axes sum to a single squared diameter, as promised.

A little more, about quadrilaterals

Before getting quite lost in why the isoperimetric problem should have exactly one solution (the answer, in brief, is relative convexity, but that's a mouthful, before we know how to spell it), a little bit about Steiner's proof, adapted slightly.

Now, last time it was pointed out that rotating a figure preserves both its area and its perimeter; and this in fact remains true of separate pieces of the figure; so, for instance, pertaining to the area within the black curve below:

one may rotate the mauve or blueish pieces, preserving their area as well as the lengths of both black and red curve or line segments. This is clearly a nifty thing to consider, as it means also that if there's a quadrilateral with the same sides as the red quadrilateral, but with greater area, then aranging the mauve and green bits around that gives a new shape with the same black perimeter, and greater area (the mauve and bluey bits will be the same, and the transparent bit in the middle can be made greater).

It just so happens that the best arrangement of vertices for the red quadrilateral is (surprise!) as a cyclic quadrilateral, aka "chord" quadrilateral, which is to say, such that the vertices are all on a circle. Perhaps the other surprise is that this is always doable. There is a bit of annoyance about proving the optimality of a cyclic arrangement, in that it works out to be more natural to maximize the square of the area --- in a calculation attributed to one Bretschneider, this is a sum of two terms, one of which is determined entirely by the four side lengths, the other proportional to their product and otherwise depending only the sum of opposite vertex angles... you can see how this is getting messy, yes? The calculus is quite straight-forward, but the geometry gets to be rather icky. I think there ought to be a 4-dimensional scissors congruence proof of this fact, considering how similar it is to Heron's formula. But we mustn't jump to conclusions! (Also, such things are difficult to read).

Anyways. Suppose, then, that the curve we had wasn't a circle. Then there must be some four points of our curve that weren't cyclic. (Three points determine a circle, a fourth is either on that circle or off it). And this in turn means there was then another curve with the same perimeter, and containing greater area.

Steiner's original proof relied on a simpler case of this quadrilateral argument, specifically that of parallelograms with fixed sides. It's easier to see that the greatest such area is a rectangle; but reducing to this case uses slightly more surgical trickery beforehand. I'm undecided, just now, which approach is tidier.

A Little Bit about Isoperimetric Inequalities

So, in Crelle XVIII, Jakob Steiner published the earliest known solution of the ordinary isoperimetric problem --- that is, among measurable plane figures with fixed perimeter, find which has the greatest area. I'm sure something like this result had been known considerably earlier; Euler and later Lagrange had formulated and solved rather general extremum problems for curves, and the isoperimetric example isn't difficult to solve. Variational problems were posed by one of the Bernoulis --- most notably the brachistochrone, of which several solutions were published simultaneously (Oh! For the good old days!). My source for Steiner's priority also claims he's the first person to have formulated the problem (or at least to have published such a formulation). So I'm inclined to suspect that Newton and Leibniz and them all thought the thing so painfully obvious that they never mentioned it.

The isoperimetric problem is considerably simpler than the brachistochrone. In fact, there are a number of similar problems which have strikingly similar solutions; for instance, among rectangles having fixed perimeter, that with the greatest area is a square, which is practically a proposition in Euclid, and quickly generalizes to the elsewhere-famous Cauchy-Shwartz-Буняковский inequality. If we said "triangles" instead of "rectangles", we'd get an equliateral triangle as the solution, and if you want to keep guessing at the best answers, you'll see this post is actually a love-song to symmetry.

The isoperimetric problem itself has rotational symmetry. Since rotating a plane figure changes neither its perimeter nor its area, it should be clear that if we had any extremal isoperimetic figure, then we could get more extremal figures by rotating the first one. That is, we could get more if there were more --- if our extremal isoperimetric figure didn't have the same symmetry as the isoperimetric problem itself. But there aren't more, there's only one. And there's a good reason there's only one, but we'll get to that another time.

For now, pleasant dreams!

An opposite limit theorem

This is something that those who ought to know these things usually do know, and eventually figure out in any case. So if it isn't usually part of your work to know these things, don't fret; I really ought to have learned it much sooner!

Suppose $D$ is a category with a terminal object, say $z$, and let $F:D\to C$ be any functor. Then the natural transformation $F0 : F\to Fz$ is initial among the category of objects $x$ of $C$ with natural transformations $F\to x$.

That is all.

So, you're wasting time with the internet...

(If you're here, you must be!)

... and this Batty fellow keeps intruding with things you tried to forget that you never learned because of how terribly dull your course on matrices in linear algebra was, or is...

So, yeah, I do topology. In fact, algebraic topology. One of the things I like about algebraic topology is that sometimes you get to draw nice pictures. OK, so I can't draw worth your trouble, but it's fun anyways. Maybe Emacs or someone will make better pictures.

One more nifty thing about the pictures you can draw in topology is that they're useful for other parts of math, too, like group theory. And this is a big deal! There are some Nice and Easy-to-Describe topological spaces that "know" how to calculate things that we can't calculate. This is part of what makes algebraic topology difficult, but also part of why it's exciting. And if we're lucky, sometimes they give us help understanding the things we can calculate. In particular, being able to describe a calculation by drawing a picture can help us lots.

That's my introduction to

Apocrypha Topologica I

Dear Mathematicelli,

I should like today to introduce an apocryphal history of Topology, both as a phenomenon and as a field of mathematical study. It will necessarily be abbreviated, full of fictions, and other more innocent errors --- hence apocryphal. But it should be ordered to the truth so far as illuminating the modern study of topology itself.

Topology was imposed on the visible Creation by God at least as early as the Second Day, when He said

6 ... "Let there be a firmament made amidst the waters: and let it divide the waters from the waters." 7 And God made a firmament, and divided the waters that were under the firmament, from those that were above the firmament, and it was so.

This highlights the first introduction of a disconnected set, a "you can't get there from here" in the world: to reach the waters above from those below you must cross this firmament, whatever it may be. And this notion of separation, whether in the absolute sense of being mutually inaccessible, or the relative sense of inhabiting disjoint neighborhoods has been a puzzle and inspiration for topologists since before we even had the name topology to describe the field.

Some might argue that the temporal ordering of days already introduced an order topology (or the causal partial-order topology we learn from the Special and General Theories of Relativity). I reply that this is missing something of the point, but if you want to write your own paper on the history of Topology that's quite alright.

Other Biblical features of topological interest in denoting separation: the cherubim posted to keep Adam and Eve out of Eden; the Red Sea; the River Jordan; the Rivers Tigris and Euphrates, that separate the Land Amidst the Rivers from surrounding territories (I mean, you can get in and out of that without crossing either, but then you'd have to cross the ~40km line segment between Palu and Hantepe... a narrow road, as geography goes!)

The next great topological discoveries were knots and chains. Windows are a nifty invention too, I suppose, but knots and chains are much more tangible. The way these work, as I'm sure you've experienced, is that various strands of rope or metal, being extended in one direction, have new ways of becoming separated from conditions they might have liked to achieve: while it would take a wall to stop you, and you can walk around a rope quite happily, a string can get stuck by a rope in one perpetual wandering-around! By the by, I don't mean mechanical knots; these are fascinating, but the mechanical distinctions between various rolling hitches are much fuzzier than the topological fact that there are more ways to get tangled in a net than in a string tied in a single loop. Think of the "Gordian Knot" versus an undoubled slipknot bow as you might use on your shoes.

As another aside, there seems some discrepancy between my predecessors in mathematical apocrypha on the one hand vs. closer accounts of Alexander's encounter with Gordy's knot --- what the knot tied up, whether Alexander really sliced it in two or what, etc., are disputed points. One way or another, there were knots and mechanical facts emphasizing topological separations, and it became a point of proverbs through the march of time.

The Gordian Knot brings us to the relatively recent period and setting known as Greek Mythology, and there, for now we will interrupt our History. Next time: strings graduate from topological features to topological tools!

The Math Prof of Your Nightmares

Whence 4?

Time for some more math! At my (new! yay!) supervisor's behest I've been trying like a primordial lungfish to breath the fresh air called spectral sequences --- how they ever managed to acclimate to such thin and rarefied reference to anything tangible is beyond me; but then, the calculations themselves are thick as water to my more-terrestrial brain.

Today is not about spectral sequences, but back to geometric measure theory. Some time ago I got as far as to outline the proportion of surface areas vs. angles

$$ S(\triangle qrs):S(\mathbb{S})::\angle qrs + \angle rsq + \angle sqr - \pi:4\pi $$

for a spherical triangle $qrs$ --- although for reasons of presentation, that tale refered to the hemispheres $A,B,C$ described by the arcs $qr,$ $rs,$ $sq$ and containing the triangle $\triangle qrs$. (Do you see some ambiguity creeping into the tale? Don't worry: make your choices and then show that they aren't important!)

So, our task today is to improve the proportionality by establishing, for a unit sphere $\mathbb{S}$, an equality $S(\triangle qrs) = \angle qrs + \angle rsq + \angle sqr - \pi$, which by the complementary proportion will give $S(\mathbb{S})=4\pi$.

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.

An exercise in geometric measure theory

echo <<eof >>/dev/null

I was given a math paper to read over that includes, in its introduction, the assertion
"the length of an algebraic curve of degree $d$ in the unit disc is less than $4d$". I don't think I'd seen this before, so I've decided to try proving something like it, and got a smaller constant than $4$, but not much smaller. (If you've not seen this problem before, can you guess?!)

"Easy" Isomorphisms

So, if you read Hatcher (or anyone half-decent) on homotopy theory, you run into a long exact sequence of groups depending on an inclusion $j:U\rightarrow X$.

$\displaystyle{ \cdots \longrightarrow \pi_2(U)\longrightarrow \pi_2(X)\longrightarrow\pi_1(X|U) \longrightarrow \pi_1(U)\longrightarrow\pi_1(X)}$

where I'm indexing the relative homotopy functors $\pi_n(X|U)$ off by one from Hatcher's notation. This makes it easier to remember when they start being abelian, when they don't have a group structure anymore ... it's also handy to note that $\pi_n(X|U)$ actually is $pi_n$ of a functorial construct on the inclusion $j$.

Anyways, we have this long exact sequence, and it comes in handy whenever $U$ (or $X$) and $(X|U)$ --- whatever that means --- have easy homotopy groups. For instance, the 2-sphere and the 1-sphere both have easy homotopy groups; it just so happens that a long slog of an argument gives us that any Jordan Curve is a topological 1-sphere. Specializing the long exact sequence to a Jordan curve $J$ in a 2-sphere, we find

$\displaystyle{ \cdots 0 \longrightarrow \mathbb{Z} \longrightarrow \pi_1(S^2|J) \longrightarrow \mathbb{Z} \longrightarrow 0 }$

at the end of our sequence. Now, from algebra, there are only two possible groups that can sit in for $\pi_1(S^2|J)$; one of them is abelian, one has an abelian subgroup of index $2$; in any case, the long exact sequence says that $\pi_1(S^2|J)$ has an element $x$ represented by a disc whose boundary is mapped to the Jordan curve $J$ --- that's surjectivity of the last nonzero map --- and has another generator $y$ mapping the boundary of the disc to a single point --- that's by injectivity of the first nonzero map. We may just as well instead take as generators $x,xy$, so they both wrap the disk's edge around $J$ once.

Now, We're not quite at the Jordan curve theorem yet, but we're pretty-darn close!

Fools seldom differ

Dear John Forbes Nash Junior,

I was thinking about isometric embedding, and on reading an interview of Mikhail Gromov decided to look up your papers on the subject.

"G-is-for-Genius Gnash", you stole my idea! and 28 years before I was born, too... "spiralling" perturbation indeed! (Goodness, this theorem is still more than twice as old as I am ...)

OK, you probably did it better than I would have. Whilst I go sulk and bask all at once, I shall also ponder Pontryagin classes, because I still don't quite get the point, exactly.

still looking for a supervisor

An apeal for clarity

Dear Fellow Mathematicians

Letters make Lousy Names For Theories! I mean, no-one goes around talking about "H-theory", it's Homology and Homological Algebra; analysis isn't "ε-δ-theory" even if within analysis "ε-δ-arguments" are a hugely fundamental skill to develop.

So, then, why have we got "q-deformation", "K-theory", or "π-categories", or "L-series"? (Unless π is some special value of n?) For one thing, when you hear it, how do you know that "q-deformation" isn't Q-deformation? Sure, "λ-calculus" is pretty special, but---accepting the Church-Turing [hypo]thesis---you might also call it "recursion".

On the one hand, "q" "K" "π" "L" and "λ" don't actually convey anything about what they signify. Within the cultures that give rise to these names, "q" is a number which might be prime, or might be close to zero, or close to 1; K is a functor to graded modules; "λ" is a piece of syntax, and could formally be replaced by "[" without loss of legibility. I don't even know what "L" and "π" are; though I do know many things that get called "L-series", and have my own ideas about what common ground they inhabit. And that brings up another thing: when the name for a province of research is an "in"-joke or otherwise obscure reference, it makes it intimidating for neighbors to take an interest in what you're really studying, and it's not clear to me that this helps you write grant proposals. Short, cultural nicknames are suitable for those intimately familiar with the subject; there should be better language for talking with foreigners.

On the other hand, there just AREN'T enough letters to reserve them for broad concepts at whim or out of laziness. So, if you're on the cusp of formalizing a grand new scheme of gadgets or tying together some ring of concepts, try to at least give them a decent, pronounceable, memorable name? It'll be very confusing when we have to re-name half of our things four decades from now, so be considerate.

From a crowded address space