Also available at http://math.ucr.edu/home/baez/week262.html
April 5, 2008
This Week's Finds in Mathematical Physics (Week 263)
John Baez
Enough nebulae! Today's astronomy picture is Saturn's moon
Titan, photographed by the Cassini probe. Red and green
represent methane absorption bands, while blue represents
ultraviolet. Note the incredibly deep atmosphere - hundreds
of kilometers deep. That's because Titan has a dense atmosphere
but not much gravity. The pale feature in the center here
is called Xanadu.
1) Astronomy Picture of the Day, Tantalizing Titan,
http://apod.nasa.gov/apod/ap041028.html
If you fell into Titan's atmosphere, here's what you'd see.
Unlike the previous picture, this is in natural colors, taken
by the Cassini probe on March 31st, 2005 from a distance of
just 9,500 kilometers:
2) Wikipedia, Titan's atmosphere,
http://en.wikipedia.org/wiki/Titan_(moon)#Atmosphere
The orange stuff is hydrocarbon "smog", perhaps made of chemicals
called "tholins" which I don't really understand. When you got
further down the atmosphere would be so thick, and the gravity
so low, that you could fly through it by flapping wings attached
to your arms. Unfortunately the atmosphere would be very cold,
and unbreathable: mostly nitrogen, with a little methane and
ethane. (I wrote about the hydrocarbon rain on Titan back in
"week160", and showed you the first pictures of its lakes in
"week210".)
Astronomy is great, but today I want to talk about group theory.
As you may have heard, John Thompson and Jacques Tits won the
2008 Abel prize for their work on groups:
2) Abel Prize, 2008 Laureates,
http://www.abelprisen.no/en/prisvinnere/2008/
If you want a fun, nontechnical book that gives a good taste of
the sort of things Thompson thought about, try this:
3) Marcus du Sautoy, Symmetry: a Journey into the Patterns of
Nature, HarperCollins, 2008.
Mathematicians will enjoy this book for its many anecdotes about
the heroes of symmetry, from Pythagoras to Thompson and other
modern group theorists. Nonmathematicians will learn a lot about
group theory in a fun easy way.
As a PhD student working under Saunders Mac Lane, Thompson began
his career with a bang, by solving a 60-year-old conjecture posed
by the famous group theorist Frobenius.
4) Mactutor History of Mathematics Archive, John Griggs Thompson,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Thompson_John.html
But, he's mainly famous for helping prove an even harder theorem
that's even simpler to state - one of those precious nuggets of
knowledge that mathematicians fight so hard to establish:
"Every finite group with an odd number of elements is solvable."
We say a group is "solvable" if it can be built out of abelian
groups in finitely many stages: the group at each stage mod the
group at the previous stage must be abelian. The term "solvable"
comes from Galois theory, since we can solve a polynomial equation
using radicals iff its Galois group is solvable.
Way back in 1911, Burnside conjectured that every finite group
with an odd number of elements is solvable. John Thompson and
Walter Feit proved this in 1963. Their proof took all 255 pages
of an issue of the Pacific Journal of Mathematics!
The proof has been simplified a bit since then, but not much.
Versions can be found in two different books, and there is a
project underway to verify it by computer:
5) Wikipedia, Feit-Thompson Theorem,
http://en.wikipedia.org/wiki/Feit-Thompson_theorem
This theorem, also called the "odd order theorem", marked a trend
toward really long proofs in finite group theory, as part of a
quest to classify finite "simple" groups. A simple group is one
that has no nontrivial normal subgroups. In other words: there's
no way to find a smaller group inside it, mod out by that, and
get another smaller group. So, more loosely speaking, we can't
build it up in several stages: it's a single-stage affair, a basic
building block.
One reason finite simple groups are important is that *every*
finite group can be built up in stages, where the group at each
stage mod the group at the previous stage is a finite simple
group. So, the finite simple groups are like the "prime
numbers" or "atoms" of finite group theory.
The first analogy is nice because *abelian* finite simple groups
practically *are* prime numbers. More precisely, every abelian
finite simple group is Z/p, the group of integers mod p, for some
prime p. So, building a finite group from simple groups is a grand
generalization of factoring a natural number into primes.
However, the second analogy is nice because just as you can build
different molecules with the same collection of atoms, you can
build different finite groups from the same finite simple groups.
I actually find a third analogy helpful. As I hinted, for any finite
group we can find an increasing sequence of subgroups, starting with
the trivial group and working on up to the whole group, such that
each subgroup mod the previous one is some finite simple group.
So, we're building our group as a "layer-cake" with these finite
simple groups as "layers".
But: knowing the layers is not enough: each time we put on the
next layer, we also need some "frosting" or "jam" to stick it on!
Depending on what kind of frosting we use, we can get different
cakes!
To complicate the analogy, stacking the layers in different
orders can sometimes give the same cake. This is reminiscent
of how multiplying prime numbers in different orders gives the
same answer. But, unlike multiplying primes, we can't *always*
build our layer cake in any order we like.
Apart from the order, though, the layers are uniquely determined -
just as every natural number has a unique prime factorization.
This fact is called the "Jordan-Holder theorem", and these layer
cakes are usually called "composition series". For more, try this:
6) Wikipedia, Composition series,
http://en.wikipedia.org/wiki/Composition_series
But let's see some examples!
Suppose we want to build a group out of just two layers, where
each layer is the group of integers mod 3, otherwise known as Z/3.
There are two ways to do this. One gives Z/3 + Z/3, the group
of pairs of integers mod 3. The other gives Z/9, the group of
integers mod 9.
We can think of Z/3 + Z/3 as consisting of pairs of digits 0,1,2
where we add each digit separately mod 3. For example:
01 + 02 = 00
12 + 11 = 20
11 + 20 = 01
We can think of Z/9 as consisting of pairs of digits 0,1,2 where
we add each digit mod 3, but then carry a 1 from the 1's place to
the 10's place when the sum of the digits in the 1's place exceeds
2 - just like you'd do when adding in base 3. I hope you remember
your early math teachers saying "don't forget to carry a 1!" It's
like that. For example:
01 + 02 = 10
12 + 11 = 00
11 + 20 = 01
So, the "frosting" or "jam" that we use to stick our two copies
of Z/3 together is the way we carry some information from one
to the other when adding! If we do it trivially, not carrying
at all, we get Z/3 + Z/3. If we do it in a more interesting way
we get Z/9.
In fact, this how it always works when we build a layer cake
of groups. The frosting at each stage tells us how to "carry"
when we add. Suppose at some stage we've got some group G. Then
we want to stick on another layer, some group H. An element of
the resulting bigger group is just a pair (g,h). But we add these
pairs like this:
(g,h) + (g',h') = (g + g' + c(h,h'), h + h')
where
c: H x H -> G
tells us how to "carry" from the "H place" to the "G place"
when we add. So, information percolates down when we add
two guys in the new top layer of our group.
Of course, not any function c will give us a group: we need
the group laws to hold, like the associative law. To make
these hold, the function c needs to satisfy some equations.
If it does, we call it a "2-cocycle".
These cocycles are studied in a subject called "group cohomology".
Usually people focus on the simplest case, when our original
group G is abelian, and its elements commute with everything
in the big new group we're building. If this isn't true, we
need something more general: *nonabelian* group cohomology,
often called "Schreier theory" (see "week233").
I like this layer cake business because it's charming and it
generalizes in two nice ways. First of all, it works for
lots of algebraic gadgets besides groups. Second of all, it
works for *categorified* versions of these gadgets.
For example, a group is a category with one object, all of
whose morphisms are invertible. Similarly, an "n-group" is
an n-category with one object, all of whose 1-morphisms,
2-morphisms and so on are invertible. We can build up n-groups
as layer cakes where the layers are groups. It's a more
elaborate version of what I just described - and it uses not
just "2-cocycles" but also "3-cocycles" and so on. I never
really understand group cohomology until I learned to see it
this way.
But what's *really* cool is that n-groups can also be thought of
as topological spaces. This lets us build every space as a "layer
cake" where the layers are groups! These groups are called
the "homotopy groups" of the space. The nth homotopy group keeps
track of how many n-dimensional holes the space has - see "week102"
for details.
But of course, they don't call the process of sticking these groups
together a "layer cake": that would be too undignified. They call
it a "Postnikov tower". And instead of "frosting", they speak of
"Postnikov invariants". Every space is the union of a bunch of
connected pieces, each of which is determined by its homotopy groups
and its Postnikov invariants.
(At least this is true if you count spaces as the same when they're
"weakly homotopy equivalent". This is a fairly sloppy equivalence
relation beloved by homotopy theorists. You've probably heard
how a topologist is someone who can't tell the difference between
a doughnut and a coffee cup. Actually they can tell: they just
don't care! A homotopy theorist is a more relaxed sort of guy
who doesn't even care about the difference between a doughnut
and a Moebius strip. They're both just fattened up versions of a
circle.)
Mike Shulman and I tried to explain this layer cake business here:
7) John Baez and Michael Shulman, Lectures on n-Categories
and Cohomology, to appear in n-Categories: Foundations and
Applications, eds. John Baez and Peter May. Also available
as arXiv:0608.5420.
Whoops! I see I've drifted from my supposed topic - the work of
John Thompson - to something I actually understand. It was a
digression, but not a completely pointless one. From what I've
told you, it follows that every space with finite homotopy groups
can be built as a fancy "layer cake" made of finite simple groups.
And even better, the finite simple groups have now been classified! -
we think. There are 18 infinite series of these groups, and also
26 exceptions called "sporadic" groups, ranging in size from the
five Mathieu groups (see "week234") on up to the Monster (see
"week20" and "week66").
8) Wikipedia, List of finite simple groups,
http://en.wikipedia.org/wiki/List_of_finite_simple_groups
Proving that these are all the possibilities took mathematicians
about 10,000 pages of work! The Feit-Thompson theorem is a small
but crucial piece in this enormous pyramid of proofs. There could
still be some mistakes here and there, but experts are busy working
through the details more carefully.
Among the 26 sporadic groups, one is called the Thompson group.
It was discovered by Thompson, and it's a subgroup of a version
of the group E8 defined over F_3, the field with 3 elements.
It has about 9 x 10^{16} elements, and it has a 248-dimensional
representation over F_3. I don't know much about it. I mention
it just to show what crazy possibilities had to be considered to
classify all finite simple groups - and how deeply Thompson was
involved in this work.
But what about Jacques Tits?
9) Mactutor History of Mathematics Archive, Jacques Tits
http://www-history.mcs.st-andrews.ac.uk/Biographies/Tits.html
He's not mentioned in du Sautoy's book "Symmetry", which is a
pity, but not surprising, since too many mathematicians have
studied group theory to fit comfortably in one story. He
has a sporadic finite simple group named after him, but his
work leaned in a different direction, more focused on the role
of groups in geometry. He was an honorary member of Bourbaki,
and in that role he helped awaken interest in the work of Coxeter.
I've mentioned his work on the "magic square" of exceptional
Lie groups in "week145" and "week253"... but he's more famous for
his work on "buildings", sometimes called "Bruhat-Tits buildings".
The subject of buildings has a reputation for being intimidating,
perhaps because the *definition* of a building looks scary and
unmotivated. You can read these and decide for yourself:
10) Wikipedia, Building (mathematics),
http://en.wikipedia.org/wiki/Building_%28mathematics%29
11) Kenneth S. Brown, What is a building?, Notices AMS, 49
(2002), 1244-1245. Also available at
http://www.ams.org/notices/200210/what-is.pdf
12) Paul Garrett, Buildings and Classical Groups, CRC Press,
1997. Preliminary version available at
http://www.math.umn.edu/~garrett/m/buildings/
13) Kenneth S. Brown, Buildings, Springer, 1989.
14) Mark Ronan, Lectures on Buildings, Academic Press, 1989.
Personally I found it a lot easier to start with *examples*.
So, start with any "finite reflection group" - a finite group
of transformations of R^n that's generated by reflections.
The possibilities have been completely worked out, and I listed
them back in "week62". But let's do an easy one: the symmetry
group of an equilateral triangle.
I can't resist mentioning that this group is also S_3, the group
of all permutations of the three vertices of the triangle. In fact,
this group was the star of "week261", where it showed up as the
Galois group of the cubic equation! We can solve a cubic using
radicals since this group is solvable. In other words, we can
build this group as a "layer cake" from the abelian groups Z/3 and
Z/2. The bottom layer is Z/3, the subgroup of even permutations.
The top layer is S_3 modulo the even permutations, namely Z/2.
Galois theory says you can solve a cubic by messing around a
bit, then taking a square root, and then taking a cube root.
Why a square root *first*? Because you build this sort of
layer cake from the bottom up, but you eat it from the top down,
slicing off one layer at a time.
But now we want to think about how this group is generated by
reflections. You can use just two, for example the reflections
across the mirrors labelled r and s here:
s
\ /
\ /
\ /
\ /
--------o--------r
/ \
/ \
/ \
/ \
Let's call these reflections r and s. They clearly satisfy
r^2 = s^2 = 1
but since the mirrors are at an angle of pi/3 from each other,
they also satisfy
(rs)^3 = 1
This gives a presesentation of our group S_3. We can summarize
this presentation with a little "Coxeter diagram":
3
r-------s
where the dots r and s are the generators, and the edge labelled
"3" is the interesting relation (rs)^3 = 1. I explained these
diagrams more carefully back in "week62". If you know about
Dynkin diagrams, these are pretty similar - see "week63" and "week64"
for details.
Note that the mirrors in this picture:
s
\ /
\ /
\ /
\ /
--------o--------r
/ \
/ \
/ \
/ \
chop the plane into 6 "chambers", and the group S_3 has 6 elements.
This is no coincidence - it works like this for any finite reflection
group! We can pick any chamber as our favorite and label it 1:
s
\ /
\ /
\ / 1
\ /
--------o--------r
/ \
/ \
/ \
/ \
Then, we can label any other chamber by the unique element of
our group that carries our favorite chamber to that one:
s
\ /
\ s /
sr \ / 1
\ /
--------o--------r
/ \
rsr = srs / \ r
/ rs \
/ \
If we start with chamber 1 and keep reflecting across mirrors,
we keep getting products of more and more generators until we
reach the diametrically opposite chamber, which corresponds to
the so-called "long word" in our finite reflection group. In
this case, the long word is rsr = srs.
(Fanatical devotees will also note that this equation is the
"Yang-Baxter equation" mentioned in "week261".)
Now, Coxeter thought about all this stuff, and he realized
that it was nice to introduce a polytope with one face for
each chamber - in this case, just a hexagon:
s
o-----o
rs / \1
/ \
o o
\ /
rsr \ /r
o-----o
sr
This is called the "Coxeter complex" of our finite reflection
group. Our finite reflection group acts on it, and it acts
on the faces in a free and transitive way.
But, you'll note we started with the symmetry group of an
equilateral triangle, and wound up with a hexagon! What happened?
The quick way to say it is this: combinatorially speaking,
the hexagon is the "barycentric subdivision" of our original
triangle. Not the inside of the triangle - just its surface,
or boundary! The boundary of the triangle is a simplicial
complex made of 3 vertices and 3 edges:
o
. .
. .
. o
. .
. .
o
so if we barycentrically subdivide it, we get 6 vertices
and 6 edges:
o-----o
/ \
/ \
o o
\ /
\ /
o-----o
and that's our hexagon - drawn puffed out a bit, just for the sake
of prettiness.
If this seems bizarre - and it probably does, given how lousy these
pictures are - I urge you to try the next example on your own. Take
the symmetry group of the regular tetrahedron, also known as S_4,
the group of permutations of 4 things. Show it's generated by
three reflections r,s,t with relations
r^2 = s^2 = t^2 = 1
(rs)^3 = (st)^3 = 1
rt = tr
We can summarize these with the following Coxeter diagram:
3 3
r-------s-------t
Draw all mirrors corresponding to reflections in S_4, and show
they chop 3d space into 24 chambers, one for each element of S_4.
Then, barycentrically subdivide the boundary of the tetrahedron
and check that the resulting "Coxeter complex" has 24 faces, one
inside each chamber.
Anyway, one thing Tits did is realize how these Coxeter complexes
show up in the geometry of the *Lie groups*, or more generally
*algebraic groups*, associated to Dynkin diagrams.
For example, if I take this guy:
3
r-------s
and remove some of the labels, I get the so-called A2 Dynkin diagram:
o-------o
which corresponds to the Lie group PSL(3). And, this is the group
of symmetries of projective plane geometry! Each dot in the Dynkin
corresponds to a "type of figure":
POINT LINE
o-------o
and the edge corresponds to an "incidence relation": in projective
plane geometry, a point can lie on a line. This shape, which we've
seen before:
o
. .
. .
. o
. .
. .
o
is then revealed to stand for a configuration of 3 points and 3
lines, satisfying incidence relations obvious from the picture.
To put points and lines on an equal footing, we switch to the
the Coxeter complex:
POINT LINE
o-----o
/ \
/ \
LINE o oPOINT
\ /
\ /
o-----o
POINT LINE
where now the vertices represent "figures" and the edges represent
"incidence relations". It turns out that inside any projective
plane, we can find lots of configurations like this: 3 points and
3 lines, each pair of points lying on one of the lines, and each
pair of lines intersecting in a point. Such a configuration is
called an "apartment".
If we take all the apartments coming from a projective plane, they
form a simplicial complex called a "building". And this generalizes
to any geometry corresponding to any sort of Dynkin diagram. The
building knows everything about the geometry: all the figures, all
the incidence relations.
And that's all I have time for now, but it's just the beginning of
the marvelous theory Jacques Tits worked out.
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Addenda: At least on Titan, tholins seem to be a complex brew of
compounds made by irradiation of molecular nitrogen and methane in the
upper atmosphere. The same sort of compounds could be an early
chemical step in the formation of life on Earth - that's one reason
I'm interested. They're related to PAHs, or "polycyclic aromatic
hydrocarbons", which are ubiquitous in outer space - I wrote about
those back in "week258". I guess the main difference is that tholins
contain nitrogen!
I found some more information on tholins here:
16) J. H. Waite, Jr., et al, The process of tholin formation in
Titan's upper atmosphere, Science 316 (2007), 870-875.
You can see more discussion and also questions I'm dying to know
the answers to over at the n-Category Cafe:
http://golem.ph.utexas.edu/category/2008/04/this_weeks_finds_in_mathematic_24.html
Whenever I write This Week's Finds, I come up with lots of questions. If
you can help me with some of these, I'll be really grateful.
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Quote of the Week:
It was technical - there was no way to avoid it. But it was a
wonderful thing. We'd finally busted it. But then, just before
we were about to submit the paper, Walter noticed a mistake.
- John Thompson
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