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Since this is the first Week of the new millennium, I'd like to start with a peek into the future. Not just the next hundred or thousand years, either - I'm sick of short-term planning. No, I'd like to talk about the next few billion years.
As you've probably all heard, if we don't do anything about it, the Sun will turn into a red giant in about 5 billion years. If we get our act together, we should have plenty of time to deal with this problem. But when planning for the far future, it's dangerous to be too parochial! Events outside our solar system can also affect us. For example, a nearby supernova could be a real bummer. It wouldn't be the first time: it seems that about 340,000 years ago there was one only 180 lightyears away. At this distance it would have been as bright as a full moon, and its X-rays and γ rays would have stripped off the Earth's ozone layer pretty badly for a while. A closer one could be a lot worse.
How we do know about this supernova? It's an interesting story. We happen to live in a region of space called the Local Bubble, about 300 lightyears across, in which the interstellar gas is hotter and 5 to 10 times less dense than the surrounding stuff. People wondered about the origin of this bubble until they studied a pulsar called Geminga about 300 lightyears away from us. Pulsars are rapidly spinning neutron stars formed by supernovae, and by studying their spin rate and the rate their spin is slowing down, you can guess when they were formed. Geminga turns out to be about 340,000 years old. It's moving away from us at a known rate, so back then it would have been 180 lightyears away - in just about the right place for a supernova to have created a shock wave forming the Local Bubble.
I don't know the best place to read about the Local Bubble, but this sounds promising:
1) M. J. Freyberg and J. Trumper, eds., The Local Bubble and Beyond, proceedings of the IAU Colloquium no. 166, Springer Lecture Notes in Physics 506, Springer-Verlag, Berlin, 1998.
Looking further afield, we should also watch out for the health of the Milky Way as a whole:
2) Robert Irion, A crushing end for our galaxy, Science 287 (2000), 62-64.
3) Roland Buser, The formation and early evolution of the Milky Way galaxy, Science 287 (2000), 69-74.
It now appears that the Milky Way, like most big spiral galaxies, was built up by a gradual merger of smaller clouds of stars and gas. And it seems this process is not finished. In 1994 people found a small galaxy orbiting the Milky Way, almost hidden behind the dense dust clouds in the galactic center. Called the Sagittarius dwarf galaxy, it is only about 1/1000th the mass of the Milky Way. Its eccentric orbit about our galaxy is strewn with stars pulled away from it by tidal forces, and it may have already passed through the outer parts of our galaxy's disk several times. It may not survive the next pass, due in about 750 million years.
But that's not all. The Large and Small Magellanic Clouds, visible to the naked eye in the Southern Hemisphere, are also dwarf galaxies orbiting ours. They are considerably larger than the Sagittarius dwarf galaxy. And they're not just orbiting the Milky Way: they are gradually spiralling in and getting torn apart. If nothing interrupts this process, they'll crash into our galaxy in about 10 billion years. When when this happens, the shock waves from colliding gas should create enough new stars to make our galaxy shine about 25% brighter for the next several hundred million years! This could prove quite a nuisance in these parts.
But again, we should not make the mistake of parochialism: dangers from afar may prove more urgent than those in our neighborhood. The dwarf galaxies near us are nothing compared to Andromeda. This spiral galaxy is twice the size of ours, about 2.5 million light years away, and clearly visible from the Northern Hemisphere. Unfortunately, it's also heading towards us at a speed of 140 kilometers per second! As it comes closer, gravitational attraction will speed it up, so it may hit our galaxy - or at least come close - in only 3 billion years. If this happens, the two galaxies will first whiz past or through each other, but then their gravitational attraction will pull them back together, and after 1 or 2 billion more years they should coalesce into a single big elliptical galaxy. Direct hits between individal stars are unlikely, but many existing stars will be hurled out into intergalactic space, and many new stars will be born as gas clouds collide.
You may think that I'm joking when I speak of planning ahead for such events, but I'm not. We have plenty of time, so it's not very urgent - but it's not too soon to start thinking about these things. And if you think it's hopelessly beyond our powers to deal with a collision of galaxies, please remember that 3 billion years ago we were single-celled organisms. With any luck, our abilities 3 billion years from now should compare to our present abilities as our present abilities compare to those of microorganisms! And if life on Earth screws up and dies out, well, there are plenty of other planets out there.
By the way, while we're discussing matters galactic, remember how last Week I said that the X-ray telescope Chandra has recently started taking data? Well, the interesting news is already coming in! For a long time people have wondered about the origin of the "X-ray background radiation": a diffuse X-ray glow that covers the whole sky. On Thursday, astronomers using Chandra discovered that most of this radiation actually comes from about 70 million individual point sources! Apparently, many of these are supermassive black holes at the center of galaxies. There's already a lot of evidence for such black holes - which seem to power quasars and other active galactic nuclei - but it's delightful to find them in such large numbers. It might even be taken as evidence for Smolin's hypothesis that the universe is optimized for black hole production thanks to a process of Darwinian evolution (see "week31" and "week33 for details.)
For more, try this:
4) Chandra resolves cosmic X-ray glow and finds mysterious new sources, press release available online at http://chandra.harvard.edu/press/00_releases/press_011400bg.html
You should also check out the Chandra website for nice new pictures of the black holes at the center of the Milky Way and Andromeda.
Okay... I've been sort of goofing off in the last few Weeks, but now I want to return to some hardcore mathematics. In particular, I want to talk about n-categories and homotopy theory, so I'm going to pick up "The Tale of n-Categories" roughly where I left off in "week100", and start connecting it to the little introduction to homotopy theory I gave from "week115" to "week119".
As I've said many times, the goal of n-category theory is to eliminate equations from mathematics, or at least to be able to postpone pretending that isomorphisms are equations for as long as you like. I've repeatedly described the practical benefits of this, so I won't bother doing so again - I'll assume you're convinced of it!
To achieve this goal, an n-category is supposed to be some sort of algebraic structure with objects, morphisms between objects, 2-morphisms between morphisms, and so on up to and including n-morphisms, with various ways of composing all these guys. The idea is thhen that we should never assert that two j-morphisms are equal except for j = n. Instead, we should just specify an equivalence between them. An "equivalence" is a bit like an isomorphism, but it's defined recursively from the top down. An n-morphism is an equivalence iff it's an isomorphism, that is, iff it's invertible. But for j < n, a j-morphism is an equivalence if it's invertible up to equivalence.
There are various competing definitions of n-category at present, but the key idea behind all the definitions of weak n-category is that the ways of composing j-morphisms should satisfy associativity and all the other usual laws only up to equivalence. For example, suppose we have some morphisms a: w → x, b: w → y and c: y → z in a 1-category. Then associativity holds "on the nose", i.e., as an equation:
a(bc) = (ab)c.
In a 1-category there is no opportunity for "weakening" this law. But in a weak 2-category, associativity holds only up to equivalence. In other words, we have an invertible 2-morphism called the "associator"
Aa,b,c: (ab)c => a(bc)
taking the part of the above equation.
But there's a catch: when we replace equational laws by equivalences this way, the equivalences need to satisfy laws of their own, or it becomes impossible to work with them. These laws are called "coherence laws". For the associator, the necessary coherence law is called the pentagon equation. It says that this diagram commutes:
((ab)c)d ====> (ab)(cd) ====> a(b(cd))
|| /\
|| ||
\/ ||
(a(bc))d =====================> a((bc)d)
I haven't labelled the double arrows here, but they are all 2-morphisms
built from the associator in obvious ways... obvious if you know about
2-categories, at least. The pentagon equation says that the two basic
ways of going from ((ab)c)d to a(b(cd)) by rebracketing are equal to
each other. But in fact, MacLane's "coherence theorem" says
that given the pentagon equation, you can rebracket composites of
arbitrarily many morphisms using the associator over and over to your
heart's content, and you'll never get into trouble: all the ways of
going from one bracketing to another are equal.
In a weak 3-category, the pentagon equation is replaced by a 3-morphism called the "pentagonator". This in turn satisfies a new coherence law of its own, which I can't easily draw for you, because doing so requires a 3-dimensional diagram in the shape of a polyhedron with 14 vertices, called the "associahedron".
As you might fear, this process never stops: there's an infinite list of "higher coherence laws" for associativity, which can be represented as higher-dimensional associahedra. They were discovered by James Stasheff around 1963. Here are the original papers:
5) James Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 (1963), 275-292.
James Stasheff, Homotopy associativity of H-spaces II, Trans. Amer. Math. Soc. 108 (1963), 293-312.
Personally, I find his book a lot easier to read:
6) James Stasheff, H-spaces from a Homotopy Point of View, Springer Lecture Notes in Mathematics 161, Springer-Verlag, New York, 1970.
There's a wealth of interesting combinatorics lurking in the associahedra. To my shame, I realize that I've never discussed this stuff, so I'd better say a bit about it. Then next Week I'll return to my real goal, which is to explain how you can use homotopy theory to understand coherence laws. With any luck, I'll get around to telling you all sorts of wonderful stuff about Postnikov towers, the cohomology of Eilenberg-MacLane spaces, and so on. We'll see. So much math, so little time....
Okay, here's how you build an associahedron.
First I'll describe the vertices, because they're very simple: the correspond to all the ways of bracketing a string of n letters. Well, that's a bit vague, so I'll do an example. Suppose n = 4. Then we get 5 bracketings:
((ab)c)d
(a(bc))d
(ab)(cd)
a((bc)d)
a(b(cd))
These are exactly the vertices of the pentagon I drew earlier! And this how it always works: the bracketings of n letters are the vertices of the (n-2)-dimensional associahedron. This should not be surprising, since associativity is all about bracketing.
More precisely, we're interested in the bracketings of n letters that correspond to binary planar trees with n leaves. For example, when n = 4:
a b c d
\ / / /
\/ / /
\ / / ((ab)c)d
\/ /
\ /
\/
a b c d
\ \ / /
\ \/ /
\ / / (a(bc))d
\/ /
\ /
\/
a b c d
\ / \ /
\/ \ /
\ / (ab)(cd)
\ /
\ /
\/
a b c d
\ \ / /
\ \/ /
\ \ / a((bc)d)
\ \/
\ /
\/
a b c d
\ \ \ /
\ \ \/
\ \ / a(b(cd))
\ \/
\ /
\/
We can think of these trees as recording the process of multiplying
n things, with time marching down the page.
How many binary planar trees with n leaves are there, anyway? Well, the answer is called the (n-1)st Catalan number. These numbers were first discovered by Euler, but they're named after Eugene Catalan, who discovered their relation to binary trees. Here they are, starting from the 0th one:
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, ...
The nth Catalan number is also the number of ways of taking a regular (n+2)-gon and chopping it into triangles by connecting the vertices by line segments that don't cross each other. It's also the number of ways of getting from a street corner in Manhattan to another street corner that's n blocks north and n blocks east, always driving north or east, but making sure that at no stage have you gone a greater total distance north than east. Get it? No? Maybe a picture will help! When n = 3, there are 5 ways:
. . . . . . . . . . . . . . . . . . . .
| | | | |
. . . . . . . . . . .__. . . . . . . .__.
| | | | |
. . . . . . .__. . . . . . .__.__. . .__. .
| | | | |
.__.__.__. .__.__. . .__.__. . .__. . . .__. . .
I leave it as a puzzle for you to understand why all these things are
counted by the Catalan numbers. If you want to see nicer pictures of
all these things, go here:
7) Robert M. Dickau, Catalan numbers, http://forum.swarthmore.edu/advanced/robertd/catalan.html
For more problems whose answer involves the Catalan numbers, try this:
8) Kevin Brown, The meanings of Catalan numbers, http://www.seanet.com/~ksbrown/kmath322.htm
To figure out a formula for the Catalan numbers, we can use the technique of generating functions:
9) Herbert Wilf, Generatingfunctionology, Academic Press, Boston, 1994. Also available for free at http://www.cis.upenn.edu/~wilf/
Briefly, the idea is to make up a power series T(x) where the coefficient of xn is the number of n-leaved binary trees. Since by some irritating accident of history people call this the (n-1)st Catalan number, we have:
T(x) = sum Cn-1 xn
We can do this trick whenever we're counting how many structures of some sort we can put on an n-element set. Nice operations on structures correspond to nice operations on formal power series. Using this correspondence we can figure out the function T and then do a Taylor expansion to determine the Catalan numbers. Instead of explaining the theory of how this all works, I'll just demonstrate it as a kind of magic trick.
So: what is a binary tree? It's either a binary tree with one leaf (the degenerate case) or a pair of binary trees stuck together. Now let's translate this fact into an equation:
T = x + T2
Huh? Well, in this game "plus" corresponds to "or", "times" corresponds to "and", and the power series "x" is the generating function for binary trees with one leaf. So this equation really just says "a binary tree equals a binary tree with one leaf or a binary tree and a binary tree".
Next, let's solve this equation for T. It's just a quadratic equation, so any high school student can solve it:
T = (1 - sqrt(1 - 4x))/2.
Now if we do a Taylor expansion we get
T = x + x2 + 2x3 + 5x4 + 14x5 + 42x6 + ...
Lo and behold - the Catalan numbers! If we're a bit smarter and use the binomial theorem and mess around a bit, we get a closed-form formula for the Catalan numbers:
Cn = (2n choose n)/(n + 1)
Neat, huh? If you want to understand the category-theoretic foundations of this trick, read about Joyal's concept of "species". This makes precise the notion of a "structure you can put on a finite set". For more details, try:
10) Andre Joyal, Une theorie combinatoire des series formelles, Adv. Math. 42 (1981), 1-82.
11) F. Bergeron, G. Labelle, and P. Leroux, Combinatorial species and tree-like structures, Cambridge, Cambridge U. Press, 1998.
Anyway, now we know how many vertices the associahedron has. But what about all the higher-dimensional faces of the associahedron? There's a lot to say about this, but it's basically pretty simple: all the faces of the (n-2)-dimensional associahedron correspond to planar trees with n leaves. It gets a little tricky to draw using ASCII, so I'll just do the case n = 3. The 1-dimensional associahedron is the unit interval, and in terms of trees it looks like this:
a b c a b c a b c
\ / / \ | / \ \ /
\/ / \ | / \ \/
\ / \ | / \ /
\/ \|/ \/
.............................
Over at the left end of the interval we have the binary tree
corresponding to first composing a and b, and then composing the
result with c. At the right end, we have the binary tree
corresponding to first composing b and c, and then composing the
result with a. In the middle we have a ternary tree that
corresponds to simultaneously composing a, b, and c.
Actually, we can think of any point in the (n-2)-dimensional associahedron as an n-leaved tree whose edges have certain specified lengths, so as you slide your finger across the 1-dimensional associahedron above, you can imagine the left-hand tree continuously "morphing" into the right-hand one. In this way of thinking, each point of the associahedron corresponds to a particular n-ary operation: a way of composing n things. To make this precise one must use the theory of "operads". The theory of operads is really the royal road to understanding n-categories, coherence laws, and their relation to homotopy theory.... But here, alas, I must stop.
Footnote - If you want to know more about the deep inner meaning of the Catalan numbers, try this:
12) Richard P. Stanley, Enumerative Combinatorics, volume 2, Cambridge U. Press, Cambridge, 1999, pp. 219-229.
It lists 66 different combinatorial interpretations of these numbers! As an exercise, it urges you to prove that they all work, ideally by finding 4290 "simple and elegant" bijections between the various sets being counted.
(Thanks go to my pal Bill Schmitt for mentioning this reference.)
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