# May 18, 2008 {#week264}

Here's a puzzle. Guess the next term of this sequence:

::: {align="center"}
1, 1, 2, 3, 4, 5, 6, ...
:::

and then guess the *meaning* of this sequence! I'll give away the
answer after telling you about Coleman's videos on quantum field theory
and an amazing result on the homotopy groups of spheres.

But first... the astronomy picture of the day.

The Eaton Collection at UC Riverside may be the world's best library of
science fiction:

1) The Eaton Collection of Science Fiction, Fantasy, Horror and Utopian
Literature, `http://eaton-collection.ucr.edu/`.

Right now my wife Lisa Raphals is attending a conference there on the
role of Mars in SF, called "Chronicling Mars". Gregory Benford,
Frederik Pohl, Greg Bear, David Brin, Kim Stanley Robinson and even Ray
Bradbury are all there! But for some reason I'm staying home working on
This Week's Finds. I'd say that shows true devotion - or maybe just
stupidity.

Anyway, in honor of the occasion, here's an incredible closeup of a
crater on Mars' moon Phobos:

::: {align="center"}
[![](phobos.jpg){width="500"}](http://apod.nasa.gov/apod/ap080410.html)
:::

2) Astronomy Picture of the Day, Stickney Crater
`http://apod.nasa.gov/apod/ap080410.html`

It's another great example of how machines in space now deliver many
more thrills per buck than the old-fashioned approach using canned
primates. This photo was taken by HiRISE, the High Resolution Imaging
Science Experiment - the same satellite that took the stunning photos of
Martian dunes which graced ["Week 262"](#week262).

Mars has two moons, Phobos and the even tinier Deimos. Their names mean
"fear" and "dread" in Greek, since in Greek mythology they were sons
of Mars (really Ares), the god of war.

Interestingly, Kepler predicted that Mars had two moons before they were
seen. This sounds impressive, but it was simple interpolation, since
Earth has 1 moon and Jupiter has 4. Or at least Galileo saw 4 - now we
know there are a lot more.

Phobos is only 21 kilometers across, and the big crater you see here -
Stickney Crater - is about 9 kilometers across. That's almost half the
size of the whole moon! The collision that created it must have almost
shattered Phobos.

Phobos is so light - just twice the density of water - that people once
thought it might be hollow. This now seems unlikely, though it's been
the premise of a few SF stories. It's more likely that Phobos is a
loosely packed pile of carbonaceous chondrites captured from the
asteroid belt.

Phobos orbits so close to Mars that it zips around once every 8 hours,
faster than Mars itself rotates! Oddly, in 1726 Jonathan Swift wrote
about two moons of Mars in his novel "Gulliver's Travels" - and he
guessed that the inner one orbited Mars every 10 hours.

Gravitational tidal forces are dragging Phobos down, so in only 10
million years it'll either crash or - more likely - be shattered by
tidal forces and form a ring of debris.

So, enjoy it while it lasts.

Anyone who's seriously struggled to master quantum field theory is
likely to have profited from this book:

3) Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures,
Cambridge U. Press, Cambridge, 1988.

It's brimming with wisdom and humor. You should have already
encountered quantum field theory before trying it: what you'll get are
deeper insights.

But what if you're just getting started?

Sidney Coleman, recently deceased, was one of the best quantum field
theorists from the heyday of particle physics. As a grad student I took
a course on quantum field theory from Eddie Farhi, who said he based his
class on the notes from Coleman's class at Harvard. So, I've always
been curious about these notes. Now they're available online in
handwritten form:

4) Sidney Coleman, lecture notes on quantum field theory, transcribed
by Brian Hill, `http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf` and
`http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf`

Someone should LaTeX them up!

Even more fun, you can now see *videos* of Coleman teaching quantum
field theory:

5) Sidney Coleman, Physics 253: Quantum Field Theory, 50 lectures
recorded 1975-1976, `http://www.physics.harvard.edu/about/Phys253.html`

This is a younger, hipper Coleman than I'd ever seen: long-haired,
sometimes puffing on a cigarette between sentences. He begins by saying
"Umm... this is Physics 253, a course in relativistic quantum
mechanics. My name is Sidney Coleman. The apparatus you see around you
is part of a CIA surveillance project."

I wish I'd had access to these when I was a kid!

Now for some miraculous math. Daniel Moskovich kindly pointed out a
paper that describes all the homotopy groups of the 2-sphere, and I want
to summarize the main result.

I explained the idea of homotopy groups back in
["Week 102"](#week102). Very roughly, the $n$th homotopy group of a
space X, usually denoted \pi_n(X), is the set of ways you can map an
n-sphere into that space, where we count two ways as the same if you can
continuously deform one to the other. If a space has holes, homotopy
groups are one way to detect those holes.

Homotopy groups are notoriously hard to compute - so even for so humble
a space as the 2-sphere, S^2, there's a sense in which "nobody
knows" all its homotopy groups. People know the first 64, though. Here
are a few:

\pi_1(S^2) = 0\
\pi_2(S^2) = Z\
\pi_3(S^2) = Z\
\pi_4(S^2) = Z/2\
\pi _5(S^2) = Z/2\
\pi _6(S^2) = Z/4 \times Z/3\
\pi _7(S^2) = Z/2\
\pi _8(S^2) = Z/2\
\pi ~9~(S^2) = Z/3\
\pi ~10~(S^2) = Z/3 \times Z/5\
\pi ~11~(S^2) = Z/2\
\pi ~12~(S^2) = Z/2 \times Z/2\
\pi ~13~(S^2) = Z/2 \times Z/2 \times Z/3\
\pi ~14~(S^2) = Z/2 \times Z/2 \times Z/4 \times Z/3 \times Z/7\
\pi ~15~(S^2) = Z/2 \times Z/2\

Apart from the fact that they're all abelian groups, all finite except
for the first two, it's hard to spot any pattern!

In fact there's a majestic symphony of patterns in the homotopy groups
of spheres, starting from ones that are easy to explain and working on
up to those that push the frontiers of mathematics, like elliptic
cohomology. But, many of these patterns are too complex for present-day
mathematics until we use some tricks to "water down" or simplify the
homotopy groups.

So, what people often do first is take the limit of \pi ~n+k~(S^n) as n \to
\infty, getting what's called the $k$th "stable" homotopy group of spheres.
It's a wonderful but well-understood fact that these limits really
exist. But so far, even these are too complicated to understand until we
work "at a prime p". This means that we take the $k$th stable homotopy
group of spheres and see which groups of the form Z/p^n show up in it.
For example,

\pi ~14~(S^2) = Z/2 \times Z/2 \times Z/4 \times Z/3 \times Z/7

but if we work "at the prime 2" we just see the Z/2 \times Z/2 \times Z/4.

After all this data processing, we get some astounding pictures:

::: {align="center"}
[![](stable_homotopy_groups_hatcher.gif)](http://www.math.cornell.edu/~hatcher/stemfigs/stems.html)
:::

This picture summarizes the first 999 stable homotopy groups of spheres
at the prime 5. To understand exactly what it means, read this:

6) Allen Hatcher, Stable homotopy groups of spheres,
`http://www.math.cornell.edu/~hatcher/stemfigs/stems.html`

Order teetering on the brink of chaos! If you're brave, you can learn
more about this stuff here:

7) Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups of
Spheres, AMS, Providence, Rhode Island, 2003.

If you're less brave, I strongly suggest starting here:

8) Wikipedia, Homotopy groups of spheres,
`http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres`

But now, I want to talk about an amazing paper that pursues a very
different line of attack. It gives a beautiful description of *all* the
homotopy groups of S^2, in terms of braids:

9) A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations,
braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326.
Also available at `http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf`

For this you need to realize that for any n, there's a group B_n whose
elements are $n$-strand braids. For example, here's an element of B_3:

    |   |  |  
     \ /   | 
      /    |       
     / \   |
    |   \ /    
    |    /         
    |   / \   
     \ /   | 
      /    |      
     / \   |
    |   \ /    
    |    /        
    |   / \   
     \ /   | 
      /    |      
     / \   |
    |   \ /    
    |    /         
    |   / \   
    |  |   |

I actually talked about this specific braid back in
["Week 233"](#week233). But anyway, we count two braids as the same
if you can wiggle one around until it looks like the other without
moving the ends at the top and bottom - which you can think of as nailed
to the ceiling and floor.

How do braids become a group? Easy: we multiply them by putting one on
top of the other. For example, this braid:

         |   |   |
          \ /    |
    A  =   /     |
          / \    |
         |   |   |

times this one:

         |   |   |   
         |    \ /    
    B =  |     /    
         |    / \  
         |   |   |

equals this:

         |   |   |
          \ /    |
           /     |
          / \    |
         |   |   |
    AB = |   |   |   
         |    \ /    
         |     /    
         |    / \  
         |   |   |

and in fact the big one I showed you earlier is (AB)^3.

As you let your eye slide from the top to the bottom of a braid, the
strands move around. We can visualize their motion as a bunch of points
running around the plane, never bumping into each other. This gives an
interesting way to generalize the concept of a braid! Instead of points
running around the plane, we can have points running around S^2, or
some other surface X. So, for any surface X and any number n of strands,
we get a "surface braid group", called B_n(X).

As I hinted in ["Week 261"](#week261), these surface braid groups
have cool relationships to Dynkin diagrams. I urged you to read this
paper, and I'll urge you again:

10) Daniel Allcock, Braid pictures for Artin groups, available as
[`arXiv:math.GT/9907194`](http://arxiv.org/abs/math.GT/9907194).

But for now, we just need the "spherical braid group" B_n(S^2)
together with the usual braid group B_n.

Let's say a braid is "Brunnian" if when you remove any one strand,
the remaining braid becomes the identity: you can straighten out all the
remaining strands to make them vertical. It's a fun little exercise to
check that Brunnian braids form a subgroup of all braids. So, we have an
n-strand Brunnian braid group BB_n.

The same idea works for braids on other surface, like the 2-sphere. So,
we also have an $n$-strand *spherical* Brunnian braid group BB_n(S^2).

Now, there's obvious map

B_n \to B_n(S^2)

Why? An element of B_n describes the motion of a bunch of points
running around the plane, but the plane sits inside the 2-sphere: the
2-sphere is just the plane with an extra point tacked on. So, an
ordinary braid gives a spherical braid.

This map clearly sends Brunnian braids to spherical Brunnian braids, so
we get a map

f\colon BB_n \to BB_n(S^2)

And now we're ready for the shocking theorem of Berrick, Cohen, Wong
and Wu:

**Theorem:** For n > 3, \pi_n(S^2) is BB_n(S^2) modulo the image of
f.

In something more like plain English: when n is big enough, the nth
homotopy group of the 2-sphere consists of spherical Brunnian braids
modulo ordinary Brunnian braids!

Zounds! What do the homotopy groups of S^2 have to do with braids?
It's not supposed to be obvious! The proof of this result is long and
deep, making use of flows on metric spaces, and also the fact that all
the Brunnian braid groups BB_n fit together into a "simplicial group"
whose $n$th homology is the $n$th homotopy group of S^2. I'd love to
understand all this stuff, but I don't yet.

This result doesn't instantly help us "compute" the homotopy groups
of S^2 - at least not in the sense of writing them down as a product of
groups like Z/p^n. But, it gives a new view of these homotopy groups,
and there's no telling where this might lead.

When I was first getting ready to write this article, I was also going
to tell you about some amazing descriptions of the homotopy groups of
the *3-sphere*, due to Wu.

However, I later realized - first to my shock, and then my embarrassment
for not having known it already - that the $n$th homotopy group of S^3 is
*the same* as the $n$th homotopy group of S^2, at least for n > 2. Do
you see why?

Given this, it turns out that Wu's results are predecessors of the
theorem just stated, a bit more combinatorial and less "geometric".
Wu's results appeared here:

12) Jie Wu, On combinatorial descriptions of the homotopy groups of
certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513. Also
available at `http://www.math.nus.edu.sg/~matwujie/newnewpis_3.pdf`

Jie Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002),
645-662. Also available at
`http://www.math.nus.edu.sg/~matwujie/wgroup05-19-01.pdf`

and there's a nice summary of these results on his webpage:

13) Jie Wu, 2.1 Homotopy groups and braids, halfway down the page at
`http://www.math.nus.edu.sg/~matwujie/Research2.html`

See also this expository paper:

14) Fred R. Cohen and Jie Wu, On braid groups and homotopy groups,
Geometry & Topology Monographs 13 (2008), 169-193. Also available at
`http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf`

Next I want to talk about puzzle mentioned at the start of this Week's
Finds... but first I should answer the puzzle I just raised. Why do the
homotopy groups of S^2 match those of S^3 after a while? Because of
the Hopf fibration! This is a fiber bundle with S^3 as total space,
S^2 as base space and S^1 as fiber:

S^1 \to S^3 \to S^2

Like any fiber bundle, it gives a long exact sequence of homotopy groups
as explained in ["Week 151"](#week151):

... \to \pi_n(S^1) \to \pi_n(S^3) \to \pi_n(S^2) \to \pi _{n-1}(S^1) \to ...

but the homotopy groups of S^1 vanishes after the first, so we get

... \to 0 \to \pi_n(S^3) \to \pi_n(S^2) \to 0 \to ...

for n > 2, which says that

\pi_n(S^3) \cong \pi_n(S^2)

Okay, now for this mysterious sequence:

::: {align="center"}
1, 1, 2, 3, 4, 5, 6, ...
:::

The next term is obviously 7. If you guessed anything else, you were
over-analyzing. So the real question is: why the funny "hiccup" at the
beginning? Why the repeated 1?

You'll find two explanations of this sequence in Sloane's Online
Encyclopedia of Integer Sequences, but neither of them is the reason
James Dolan and I ran into it. We were learning about theta
functions...

Say you have a torus. Then the complex line bundles over it are
classified by an integer called their "first Chern number". In some
sense, this integer this measures how "twisted" the bundle is. For
example, you can put any connection on the bundle, compute its curvature
2-form, and integrate it over the torus: up to some constant factor,
you'll get the first Chern number.

A torus is a $2$-dimensional manifold, but we can also make it into a
1-dimensional *complex* manifold, often called an "elliptic curve". In
fact we can do this in infinitely many fundamentally different ways, one
for each point in the "moduli space of elliptic curves". I've
explained this repeatedly here - try ["Week 125"](#week125) for a
good starting-point - so I won't do so again. The details don't really
matter here.

Back to line bundles. If we pick an elliptic curve, we can try to
classify the *holomorphic* complex line bundles over it - that is, those
where the transition functions are holomorphic (or in other words,
complex-analytic). Here the classification is subtler. It turns out you
need, not just the first Chern number, which is discrete, but another
parameter which can vary in a *continuous* way.

Interestingly, this other parameter can be thought of as just a point on
your elliptic curve! So, an elliptic curve is a space that classifies
holomorphic line bundles over itself - at least, those with fixed first
Chern number. Curiously circular, eh? This is just one of several
curiously circular classification theorems that happen in this game...

But I'm actually digressing a bit - I'm having trouble resisting the
temptation to explain everything I've just been learning, since it's
so simple and beautiful. Don't worry - all you need to know is that
holomorphic line bundles over an elliptic curve are classified by an
integer and some other continuous parameter.

The puzzle then arises: how many holomorphic sections do these line
bundles have? More precisely: what's the *dimension* of the space of
holomorphic sections?

Before I answer this, I can't resist adding that these holomorphic
sections have a long and illustrious history - they're called "theta
functions", and you can learn about them here:

15) Jun-ichi Igusa, Theta Functions, Springer, Berlin, 1972.

16) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser,
Boston, 1983-1991.

They're important in geometric quantization, where holomorphic sections
of line bundles describe states of quantum systems, and the reciprocal
of the first Chern number is proportional to Planck's constant. In
fact, I first ran into theta functions years ago, when trying to
quantize a black hole - see the end of ["Week 112"](#week112) for
more details.

But anyway, here's the answer to the puzzle. The dimension turns out
not to depend on the continuous parameter labelling our line bundle, but
only on its first Chern number. If that number is negative, the
dimension is 0. But if it's 0,1,2,3,4,5,6 and so on, the dimension goes
like this:

::: {align="center"}
1, 1, 2, 3, 4, 5, 6,...
:::

Now, this sequence is fairly weird, because of the extra "1" at the
beginning. I hadn't noticed this back when I was quantizing black
holes, because the extra "1" happens for first Chern number zero,
which would correspond to Planck's constant being *infinite*. But now
that I'm just thinking about math, it sticks out like a sore thumb!

It's got to be right, since the line bundle with first Chern number
zero is the trivial bundle, its sections are just functions, and the
only holomorphic functions on a compact complex manifold are constants -
so there's a $1$-dimensional space of them. But, it's weird.

Luckily, Jim figured out the explanation for this sequence. First of
all, we can encode it into a power series:

    1 + x + 2x2 + 3x3 + 4x4 + ...

which we can rewrite as a rational function:

                                           (1-x6)
    1 + x + 2x2 + 3x3 + 4x4 + ...  =  ------------------
                                      (1-x)(1-x2)(1-x3)

Now, the reason for doing this is that we can pick a line bundle of
first Chern number 1, say L, and get a line bundle of any Chern number n
by taking the $n$th tensor power of L - let's call that L^\otimesn^. We can
multiply a section of L^\otimesn^ and a section of L^\otimesm^ to get a section of
L^\otimes(n+m)^. So, all these spaces of sections we're studying fit together
to form a commutative graded ring! And, whenever you have a graded ring,
it's a good idea to write down a power series that encodes the
dimensions of each grade, just as we've done above. This is called a
"Poincare series".

And, when you have a commutative graded ring with one generator of
degree 1, one generator of degree 2, one generator of degree 3, one
relation of degree 6, and no "relations between relations" (or
"syzygies"), its Poincare series will be

            (1-x6)
      ------------------
      (1-x1)(1-x2)(1-x3)

That's how it always works - think about it.

So, it's natural to hope that our ring built from holomorphic sections
of all the line bundles L^\otimesn^ will have one generator of degree 1, one
of degree 2, one of degree 3, and one relation of degree 6.

And, this seems to be true!

As I mentioned, people usually call these holomorphic sections "theta
functions". So, what we're getting is a description of the ring of
theta functions in terms of generators and relations.

How does it work, exactly? Well, I must admit I'm not quite sure. Jim
has some ideas, but it seems I need to do something a bit different to
get his story to work for me. Maybe it goes something like this. We can
write any elliptic curve as the solutions of this equation:

y^2 = x^3 + Bx + C

for certain constants B and C that depend on the elliptic curve. (See
["Week 13"](#week13) and ["Week 261"](#week261) for details.)
Now, this equation is not homogeneous in the variables y and x, but we
can think of it as homogeneous in a sneaky sense if we throw in an extra
variable like this:

y^2 = x^3 + Bxz^4 + Cz^6

and decree that:

y has grade 3\
x has grade 2\
z has grade 1\

Then all the terms in the equation have grade 6. So, we're getting a
commutative graded ring with generators of degree 1, 2, and 3 and a
relation of grade 6. And, I'm hoping this ring consists of algebraic
functions on the total space of some line bundle L* over our elliptic
curve. z should be a function that's linear in the fiber directions,
hence a section of L. x should be quadratic in the fiber directions,
hence a section of L^\otimes2^. And y should be cubic, hence a section of
L^\otimes3^. If L has first Chern number 1, I think we're in business.

If anybody knows about this stuff, I'd appreciate corrections or
references.

There's a *lot* more to say about this business... because it's all
part of a big story about elliptic curves, theta functions and modular
forms. But, I want to quit here for now.

------------------------------------------------------------------------

**Addenda:** I thank David Corfield for pointing out how to get ahold of
Wu's papers free online - and earlier, for telling me Wu's
combinatorial description of \pi_3(S^2).

Martin Ouwehand told me that some of Coleman's lecture notes on quantum
field theory are available in TeX here:

17) Sidney Coleman, Quantum Field Theory, first 11 lectures notes TeXed
by Bryan Gin-ge Chen, available at
`http://www.physics.upenn.edu/~chb/phys253a/coleman/`

James Dolan pointed out that this article:

18) Wikipedia, Riemann-Roch theorem,
`http://en.wikipedia.org/wiki/Riemann-Roch`

has some very relevant information on the sequence

::: {align="center"}
1, 1, 2, 3, 4, 5, 6, ...
:::

though it's phrased not in terms of "sections of line bundles", but
instead in terms of "divisors" (secretly another way of talking about
the same thing). Let me quote a portion, just to whet your interest:

> We start with a connected compact Riemann surface of genus g, and a
> fixed point P on it. We may look at functions having a pole only at P.
> There is an increasing sequence of vector spaces: functions with no
> poles (i.e., constant functions), functions allowed at most a simple
> pole at P, functions allowed at most a double pole at P, a triple
> pole, ... These spaces are all finite dimensional. In case g = 0 we
> can see that the sequence of dimensions starts
>
> ::: {align="center"}
> 1, 2, 3, ...
> :::
>
> This can be read off from the theory of partial fractions. Conversely
> if this sequence starts
>
> ::: {align="center"}
> 1, 2, ...
> :::
>
> then g must be zero (the so-called Riemann sphere).
>
> In the theory of elliptic functions it is shown that for g = 1 this
> sequence is
>
> ::: {align="center"}
> 1, 1, 2, 3, 4, 5 ...
> :::
>
> and this characterises the case g = 1. For g > 2 there is no set
> initial segment; but we can say what the tail of the sequence must be.
> We can also see why g = 2 is somewhat special.
>
> The reason that the results take the form they do goes back to the
> formulation (Roch's part) of the \[Riemann-Roch\] theorem: as a
> difference of two such dimensions. When one of those can be set to
> zero, we get an exact formula, which is linear in the genus and the
> degree (i.e. number of degrees of freedom). Already the examples given
> allow a reconstruction in the shape
>
> ::: {align="center"}
> dimension - correction = degree - g + 1.
> :::
>
> For g = 1 the correction is 1 for degree 0; and otherwise 0. The full
> theorem explains the correction as the dimension associated to a
> further, 'complementary' space of functions.

You can see more discussion of this Week's Finds at the [$n$-Category Café](http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_25.html).

------------------------------------------------------------------------

*The career of a young theoretical physicist consists of treating the
harmonic oscillator in ever-increasing levels of abstraction.* - Sidney
Coleman