# November 28, 2008 {#week272}

Today I want to talk about the Enceladus plumes, the Io flux tube,
special properties of the number 6 - and the wonders of standard Borel
spaces, commutative von Neumann algebras and Polish groups.

Let's start by leaving the fire and brimstone of Io. Let's sail out to
the icy splendor of Saturn's moon Enceladus, and gaze at its geysers:

::: {align="center"}
![](enceladus_geysers.jpg)
:::

1) NASA Photojournal, Jet blue,
`http://photojournal.jpl.nasa.gov/catalog/PIA08386`

We've seen Enceladus once before, in ["Week 231"](#week231). I
showed you a photo of the icy geysers near its south pole, taken by the
Cassini probe back in 2005. Above is a prettier version of the same
picture, released this August to celebrate Cassini's return to
Enceladus. You can see the geysers shooting plumes of ice into outer
space!

You see, this year NASA has been using the Cassini probe to take a
better look at large cracks on the icy surface of Enceladus. They're
called "tiger stripes", and this where the geysers are. On August
11th, the Cassini probe shot past Enceladus at a distance of just 50
kilometers. It took some wonderful photos as it approached:

::: {align="center"}
![](enceladus_tiger_stripes.jpg)
:::

2) NASA Photojournal, Great southern land,
`http://photojournal.jpl.nasa.gov/catalog/PIA11112`

Here you see the four main tiger stripes on Enceladus. They're the big
greenish cracks near the south pole, running from left to right. For
some reason these tiger stripes are named after cities mentioned in that
wonderful book of tales called One Thousand and One Nights. From top to
bottom: Damascus Sulcus, Baghdad Sulcus, Cairo Sulcus and Alexandria
Sulcus. "Sulcus" is Latin for a depression or fissure. It's often
used as a medical term, especially for features of the brain - but on
Enceladus it just means "tiger stripe".

These tiger stripes are about 2 kilometers wide and 500 meters deep.
They're flanked by ridges that are about 2-4 kilometers wide and 100
meters tall. The Damascus and Baghdad Sulci are the most active ones, as
far as geysers go:

::: {align="center"}
![](enceladus_baghdad_damascus.jpg)
:::

Alexandria Sulcus is the least active.

The white stuff in this false-color image is fine-grained ice. The green
stuff is bigger chunks of ice - even house-sized boulders. These bigger
chunks are concentrated along valley floors and walls, and along the
upraised flanks of the tiger stripes.

At its closest approach, Cassini took much higher resolution photographs
of the tiger stripes, like this closeup of the big fork in Damascus
Sulcus:

::: {align="center"}
![](enceladus_damascus_sulcus.jpg){width="500"}
:::

3) NASA Photojournal, Damascus Sulcus on Enceladus,
`http://photojournal.jpl.nasa.gov/catalog/PIA11113`

The hard part was compensating for the fact that the spacecraft was
whizzing by the moon. The NASA team used a clever maneuver they called a
"skeet shoot" to tackle this problem:

> On Earth, skeet shooting is an outdoor shotgun sport that simulates
> shooting game birds in flight. A small Frisbee-like ceramic disk,
> called a clay pigeon, is launched through the air, usually diagonally
> across in front of a shooter armed with a shotgun. The skill in
> successfully hitting the moving clay target with the birdshot is for
> the shooter to point a little ahead of the clay pigeon, and match its
> angular velocity when the trigger is pulled. The clay pigeon then
> passes into the bird shot at exactly when the shot arrives at its
> destination in the path of the moving target. So far so good. At and
> just after closest-approach on the Enceladus 4 flyby, relative to
> Cassini and Optical Remote Sensing boresight directions, Enceladus was
> streaking too quickly across the sky for the spacecraft to be able to
> stably target and track any geological feature on the surface.
> Borrowing from the firearms sport, the trick was to turn the
> spacecraft as fast as possible in the same direction as Enceladus'
> path across the sky. The plan was to be leading Enceladus and match
> its angular velocity at the exact times when our targets of interest
> passed into our camera's field of view.

It worked amazingly well. They got some photos that could see features
as small as 10 meters! It's great fun to read blogs of the scientists
involved, and watch a video of the maneuver:

4) NASA Blogs, Enceladus - Aug08,
`http://blogs.nasa.gov/cm/blog/cassini-aug08/`

During this flyby, an infrared spectrometer on the Cassini probe saw
that the tiger stripes are significantly warmer than their surroundings.
Here's a temperature map with approximate locations of active geysers.
From left to right you can see Damascus Sulcus, Baghdad Sulcus, Cairo
Sulcus and Alexandria Sulcus:

::: {align="center"}
![](enceladus_jet_spots.jpg){width="400"}
:::

5) NASA Photojournal, Jet spots in tiger stripes,
`http://photojournal.jpl.nasa.gov/catalog/PIA10361`

Other instruments detected a plume of water vapor, ice, methane, carbon
dioxide, nitrogen, and more complicated organic compounds. All this is
good evidence for "cryovolcanism": a version of volcanic activity that
can happen on really cold worlds, with water playing the role of lava.
Cryovolcanism has also been seen on other moons of outer planets:
Triton, Europa, Titan, Ganymede and Miranda.

Cassini made two more flybys in October, and another in November:

6) NASA Blogs, Enceladus, `http://blogs.nasa.gov/cm/blog/enceladus`

And just this week, Candice Hansen and her coauthors came out with a new
paper about the geysers of Enceladus:

7) C. J. Hansen, L. W. Esposito, A. I. F. Stewart, B. Meinke, B.
Wallis, J. E. Colwell, A. R. Hendrix, K. Larsen, W. Pryor and F. Tian,
Water vapour jets inside the plume of gas leaving Enceladus, Nature 456
(27 November 2008), 477-479.

8) NASA, Enceladus Jets: Are They Wet or Just Wild?,
`http://www.nasa.gov/mission_pages/cassini/whycassini/cassinif-20081126.html`

They analyzed data from the ultraviolet spectrometer on Cassini and saw
four separate jets of water blocking out the light of distant stars.
With further clever reasoning they estimated the velocity of these jets.
They seem to be moving at supersonic speeds - up to 2000 kilometers per
hour.

Then, they argue that such high speeds are only possible if there's
lots of underground liquid water near the surface on this part of
Enceladus!

This conclusion is still controversial. Susan Kieffer of the University
of Illinois at Urbana-Champaign has a competing theory, which says the
water is in the form of gas clathrates. I showed you pictures of these
back in ["Week 269"](#week269): they're cage-shaped ice crystal
structures that trap gas molecules inside. There are about 6 trillion
tons of gas clathrates on the ocean floors here on Earth. Maybe they're
also on Enceladus:

9) Susan W. Kieffer, Xinli Lu, Craig M. Bethke, John R. Spencer,
Stephen Marshak, Alexandra Navrotsky, A clathrate reservoir hypothesis
for Enceladus' south polar plume, Science 314 (15 December 2006),
1764-1766.

Only time will tell.

What's not controversial is that the geysers of Enceladus shoot out
plumes of stuff - mainly water vapor and ice, but also other gases and
dust - at speeds exceeding escape velocity. Enceladus is small: just 500
kilometers across. So its gravity is weak; its escape velocity is low.
So it's easy for geysers to blast stuff straight out into space.

And then what? Then it spreads out and form a huge ring around Saturn:
the "E ring". The E ring is much more diffuse than the really famous
rings of Saturn. It's much further out, and usually almost invisible.
But here's a great photo of the E ring, again taken by the hero of this
story - the Cassini probe:

::: {align="center"}
![](enceladus_e_ring.jpg){width="500"}
:::

10) NASA Photojournal, Ghostly fingers of Enceladus,
`http://photojournal.jpl.nasa.gov/catalog/PIA08321`

The trick was to take the shot with the sun almost directly behind the
camera. Unexpected wispy patterns may hint at interactions between the
ring and Saturn's magnetic field. Enceladus is the white blob in the
middle of the ring. If you look carefully, you can see its geysers...
and you can even see it carving a path through the E ring!

By the way, the volcanic activity on Io also produces a kind of ring: a
of ionized sulfur, oxygen, sodium, and chlorine. And since the magnetic
field of Jupiter is very strong, and Io is close by, this causes
dramatic effects. For example, charged particles from this plasma torus
are funnelled down something called the "Io flux tube" to Jupiter's
surface, where they create light shows 1000 times brighter than the
aurora borealis here on earth! You can see pictures here:

11) Astronomy Picture of the Day, Jovian aurora,
`http://apod.nasa.gov/apod/ap980123.html`

Next: some exceptional properties of the number 6.

Every set has a group of symmetries, but so does every group. The
symmetries of a group are called "automorphisms" of the group. Every
symmetry of a set gives a symmetry of its group of its symmetries,
called an "inner automorphism". But of all finite sets, only the
six-element set has a symmetry group with an EXTRA symmetry - a symmetry
that doesn't come from a symmetry of that set! This is called an
"outer automorphism".

In more jargonesque terms: of all the permutation groups S_n, only S_6
has an outer automorphism. It has just one, and I described it here:

12) John Baez, Some thoughts on the number six,
`http://math.ucr.edu/home/baez/six.html`

Recently my friend Bruce Westbury told me some more cool facts about the
number 6. One of these is another nice description of the outer
automorphism of S_6. He heard this from the group theorist Brian
Bowditch, who heard it from Chris Penrose (Roger's son). I haven't
checked it! I'll just pass it on, as gossip, and let you see if it's
true.

It goes like this. Start with the icosahedron, drawn as a graph on the
sphere, and identify oposite points. This gives a graph on the
projective plane. This graph is called K_6: the "complete graph on 6
vertices". In other words, it's the graph with 6 vertices, each one
connected to every other one by a single edge. The faces of this graph
are triangles. Apparently there are twelve embeddings of K_6 in the
projective plane and they come in six pairs. Each permutation of the six
vertices of K_6 induces a permutation of these six pairs. This process
gives an automorphism of S_6. And apparently this is the outer
automorphism!

Westbury later pointed out this related paper:

13) Ben Howard, John Millson, Andrew Snowden, Ravi Vakil, A description
of the outer automorphism of S_6, and the invariants of six points in
projective space, available as
[`arXiv:0710.5916`](http://arxiv.org/abs/0710.5916).

I haven't checked to see if the results here imply the claim above.

Another group theorist, Derek Holt, told Westbury about an intriguing
conjecture which he believes but has not had the energy to check.

Take a finite simple group and choose two elements at random. What is
the probability that they generate the group?

One cool fact is that this probability is never zero. Every finite
simple group can be generated by two elements! So, they're all
quotients of the free group on two generators.

Another cool fact is that this probability tends to 1 as the order of
the group tends to infinity:

14) M. Liebeck and A. Shalev, The probability of generating a finite
simple group, Geometrica Dedicata 56 (1995), 103-111.

This implies that this probability attains a *mininum value* for some
finite simple group. For A_6 - the group of even permutations of a
6-element set - the probability is 53/90. Holt's conjecture is that
this is the minimum among all finite simple groups.

Here are the probabilities for a few finite simple groups, listed in
order of increasing size:

A_2                                       1                           =
1.000

A_3                                       8/9                       \~
.889

\mathrm{PSL}(2,3) \cong A_4                     2/3                       \~ .667

\mathrm{PSL}(2,4) \cong \mathrm{PSL}(2,5) \cong A_5   19/30                   \~ .633

\mathrm{PSL}(2,7) \cong \mathrm{PSL}(3,2)           19/28                   \~ .679

\mathrm{PSL}(2,8)                             71/84                   \~ .845

\mathrm{PSL}(2,9) \cong A_6                     53/90                   \~ .589

\mathrm{PSL}(2,11)                           127/165               \~ .770

A_7                                       229/315               \~ .727

\mathrm{PSL}(4,2) \cong A_8                     133/180               \~ .739

A~9~                                       15403/18144       \~ .849

Here A_n is the group of even permutations of an $n$-element set, while
\mathrm{PSL}(n,q) is the group of n \times n matrices with determinant $1$ having
entries in the field with q elements, mod multiples of the identity
matrix. Of the groups listed above, only A_4 is not simple. I included
this one just because I wanted to list *all* the unexpected isomorphisms
between "alternating" groups (A_n's) and "projective special
linear" groups (PSL's).

(Each of these isomorphisms makes a wonderful story in itself - but not
for today! If you want to know these stories, try the first of Conway's
"Three Lectures on Exceptional Groups" from his book with Sloane on
Sphere Packings, Lattices and Groups. Also try
["Week 79"](#week79), to see how these stories are related to
Galois' fatal duel and the buckyball.)

So, you see the number 6 has several special properties, all related to
permutations of a 6-element set. What do these special properties really
mean? I don't know! They're just little clues waiting for a big
mystery story to come along.

Lately I've been finishing up a big paper on infinite-dimensional
representations of 2-groups, which I'm writing along with my former
student Derek Wise, the physicist Laurent Freidel, and his student
Aristide Baratin. Derek and Aristide are doing all the really hard work,
but I used to do functional analysis when I was a youngster, and
there's a lot of measure theory in our paper, so sometimes it's my job
to dig up papers on this subject and see if they help. And this sort of
work can actually be fun, once I get into the mood.

I don't want to try to explain this paper now. I just feel like
explaining some tidbits of measure theory that I've picked up while
writing it.

A measure space is basically a space that you can do integrals over. But
some measure spaces are better than others. Some of them are immensely
huge and nasty. But "standard Borel spaces" include all the examples I
care about, and they're incredibly nice.

So, what's a standard Borel space?

It's a kind of "measurable space", meaning a space equipped a
collection of subsets that's closed under countable intersections,
countable unions and complement. Such a collection is called a
"sigma-algebra", and we call the sets in here "measurable". A
"measure" on a measurable space assigns a number between 0 and +\infty to
each measurable set, in such a way that for any countable disjoint union
of measurable sets, the measure of their union is the sum of their
measures.

A nice way to build a measurable space is to start with a topological
space. Then you take its open sets and keep taking countable
intersections, countable unions and complements until you get a
sigma-algebra. This may take a long time, but if you believe in
transfinite induction you're bound to eventually succeed. The sets in
this sigma-algebra are called "Borel sets".

A basic result in real analysis is that if you put the usual topology on
the real line, and use this to cook up a sigma-algebra as just
described, there's a unique measure on the resulting measurable space
that assigns to each interval its usual length. This is called
"Lebesgue measure".

Some topological spaces are big and nasty. But
[separable](http://en.wikipedia.org/wiki/Separable_space) [complete
metric spaces](http://en.wikipedia.org/wiki/Complete_metric_space) are
not so bad.

We don't care about the metric in this game. So, we use the term
"Polish space" for a topological space that's *homeomorphic* to a
complete separable metric space.

And often we don't even care about the topology. So, we use the term
"standard Borel space" for a measure space whose measurable sets are
the Borel sets for some topology making it into a Polish space.

In short: every complete separable metric space has a Polish space as
its underlying topological space, and every Polish space has a standard
Borel space as its underlying measurable space.

Now, it's hopeless to classify complete separable metric spaces. It's
even hopeless to classify Polish spaces. But it's not hopeless to
classify standard Borel spaces! The reason is that metric spaces are
like diamonds: you can't bend or stretch them at all without breaking
them entirely. But topological spaces are like rubber... and measurable
spaces are like dust. So, it's very hard for two metric spaces to be
isomorphic, but it's easier for their underlying topological spaces -
and even easier for their underlying measurable spaces.

For example, the line and plane are isomorphic, if we use their usual
sigma-algebras of Borel sets to make them into measurable spaces! And
the plane is isomorphic to R^n for every n, and all these are
isomorphic to a separable Hilbert space! As measurable spaces, that is.

In fact, every standard Borel space is isomorphic to one of these:

-   a countable set with its sigma-algebra of all subsets,
-   the real line with its sigma-algebra of Borel subsets.

That's pretty amazing. It means that standard Borel spaces are
classified by just their *cardinality*, which can only be finite,
countably infinite, or the cardinality of the continuum. The "continuum
hypothesis" says there's no cardinality between the countably infinite
one and the cardinality of the continuum - but we don't need the
continuum hypothesis to prove this result.

This amazing result has a nice relative in the world of von Neumann
algebras.

Starting from a measure space X, we can form the Hilbert space L^2(X).
We can also form L^\infty(X), the space of equivalence classes of bounded
measurable complex functions on X, where we identify functions that
agree except on a set of measure zero.

L^\infty(X) is an algebra where we multiply functions pointwise. We can
think of it as an algebra of multiplication operators on L^2(X). In
fact, it forms a "von Neumann algebra" - a kind of algebra that's
great for quantum theory. I defined von Neumann algebras back in
["Week 175"](#week175), so I won't do it again here.

L^\infty(X) is a very special sort of von Neumann algebra, namely a
*commutative* one. There's a nice theorem: every commutative von
Neumann algebra is isomorphic (as an abstract $C^*$-algebra) to one of
this form. Even better, every commutative von Neumann algebra of
operators on a *separable* Hilbert space is isomorphic to L^\infty(X) for
one of these choices of X:

-   a countable set (equipped with counting measure),
-   the real line (with Lebesgue measure),
-   the disjoint union of a countable set and the real line (with the
    obvious measure).

We're really talking about measure theory now, even when it seems I'm
talking about commutative von Neumann algebras. But sometimes topology
sneaks into the game. This especially true if we're studying groups. We
can talk about "standard Borel groups": standard Borel spaces made
into groups where all the group operations are measurable. But nobody
seems to know much about these! What people know about are "Polish
groups": Polish spaces made into groups where all the group operations
are continuous.

If you like keeping your categories straight, it may seem annoying to
put a topology on your group when you're trying to do measure theory.
But apparently some topology is a good thing... either that, or people
haven't found the best framework for this subject.

It's a bit slippery, you see. Not every standard Borel group comes from
a Polish group. But the counterexamples seem to be horrid. And more
importantly, every measurable homomorphism between Polish groups is
automatically continuous! So there must be some way to define Polish
groups as standard Borel groups with an extra *property* - not extra
structure.

There are other similar results with the same flavor. For example,
suppose we have a Polish group G acting as measurable transformations on
a standard Borel space X. Then we can make X into a Polish space so that
the action of G on X becomes continuous!

(Of course we also require that the new Borel sets of X are the same as
its original measurable sets.)

There are also interesting results saying that Polish spaces and Polish
groups "aren't too big". Some of these theorems mention the "Hilbert
cube" - that is, a countable product of copies of \[0,1\], with its
product topology. This space sounds big, but thanks to Tychonoff's
theorem it's compact! It's also metrizable: that is, its topology
comes from a metric. And it's separable: it has a countable dense
subset. In fact, a topological space is separable and metrizable iff
it's homeomorphic to a subset of the Hilbert cube.

Here's how the Hilbert cube relates to Polish spaces: a topological
space is a Polish space iff it's homeomorphic to a countable
intersection of open subsets of the Hilbert cube!

And here's how the Hilbert cube relates to Polish groups: a topological
group is a Polish group iff it's isomorphic to a subgroup of the group
of homeomorphisms of the Hilbert cube!

I'll wrap up with a nice fact about measures on standard Borel spaces.
A "probability measure" on a measure space X is one that assigns the
number 1 to the whole set X. Then the measure of a measurable subset S
of X can be interpreted as the probability of a point being in X. This
idea of using measures to study probabilities lies at the foundation of
modern probability theory.

But here's a lesser-known fact: the set of probability measures on X,
say M(X), is itself a measurable space!

How does this work? First we give M(X) its "weak topology". This is
the topology where a bunch of measures \mu_i converge to \mu if for every
bounded continuous function f on X,

\int f d\mu_i \to \int f d\mu

Starting with this topology and taking the Borel sets, M(X) becomes a
measurable space.

It then turns out that M(X) is a standard Borel space iff X is!

In fact, M is what category theorists call a "monad" on the category
of standard Borel spaces. I said what a monad is back in
["Week 89"](#week89), so I won't do it again, but here it
involves the presence of god-given measurable maps

M(M(X)) \to M(X)

and

X \to M(X)

The first map is the most interesting: a probability measure on the
space of probability measures of X gives a probability measure on X! In
layman's terms: if you're uncertain about how uncertain you are, you
might as well just say you're uncertain.

The second map just sends any point in X to the Dirac delta measure at
that point. In layman's terms: if you're certain, you might as well
say you're uncertain (just not very much). That is, if you're
completely certain about something, you can still describe your state of
knowledge by a probability distribution.

You can see some applications of this monad here:

15) David Corfield, Category theoretic probability theory II,
`http://golem.ph.utexas.edu/category/2007/02/category_theoretic_probability_1.html`

16) Ernst-Erich Doberkat, Characterizing the Eilenberg-Moore algebras
for a monad of stochastic relations, Universitat Dortmund, Fachbereit
Informatik, Lehrstule fuer Software-Technologie, Memo 147. Also
available at
<https://eldorado.uni-dortmund.de/bitstream/2003/2717/1/147.pdf>

For more on standard Borel spaces in probability theory, try this:

17) K. R. Parthasrathy, Probability Measures on Metric Spaces, Academic
Press, San Diego, 1967.

For standard Borel spaces and von Neumann algebras, try this:

18) W. Arveson, An Invitation to C*-Algebra, Springer, Berlin, 1976.

And for really hard-core results on standard Borel spaces and Polish
groups, try this:

19) H. Becker and A. S. Kechris, Descriptive \mathsf{Set} Theory of Polish Group
Actions, Cambridge U. Press, Cambridge, 1996.

I can't imagine any of the normal readers of This Week's Finds will
enjoy this book, but I could be wrong: there could be some really scary
people who read this column.

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

**Addenda:** I thank Greg Egan, Benoit Jubin, Ted Nitz and Ultrawaffle
for corrections and improvements. Originally I had a less complete table
of probabilities for finite simple groups, and I begged my readers for
help. Ted Nitz and Greg Egan stepped into save me.

Ted Nitz calculated the odds that a pair of elements in A_4 generates
the group and got 2/3:

>     I decided to take your challenge. For A4 the odds 
>     are 96/144 = 2/3 that a pair generates. For reference, 
>     the sage code that gave the probability is:
>
>     A4 = AlternatingGroup(4)
>     order = A4.order()
>     i = 0
>     j = 0
>     for g in A4:
>         for h in A4:
>             i += 1
>             if A4.subgroup([g,h]).order() == order:
>                 j += 1
>     print (j,i)
>     j/i
>
>     It was blindingly fast for A4, and if it ever 
>     finishes for A9, I'll let you know the probability 
>     there.
>
>     -Ted

He later realized that the A~9~ calculation would take at least a
century. Greg Egan wrote a Mathematica program based on a more efficient
algorithm and got the answer in less than a day.

For more discussion, including the idea behind Egan's algorithm, visit
the [$n$-Category Café](http://golem.ph.utexas.edu/category/2008/11/this_weeks_finds_in_mathematic_33.html).

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*The splitting into something discrete and something continuous seems to
me to be a basic issue in all morphology.* - Hermann Weyl