Also available as http://math.ucr.edu/home/baez/week257.html
October 14, 2007
This Week's Finds in Mathematical Physics (Week 257)
John Baez
Time flies! This week I'll finally finish saying what I did on
my summer vacation. After my trip to Oslo I stayed in London,
or more precisely Greenwich. While there, I talked with some good
mathematicians and physicists: in particular, Minhyong Kim, Ray
Streater, Andreas Doering and Chris Isham. I also went to a
topology conference in Sheffield... and Eugenia Cheng explained
some cool stuff on the train ride there. I want to tell you about
all this before I forget.
Also, the Tale of Groupoidification has taken a shocking new
turn: it's now becoming available as a series of *videos*.
But first, some miscellaneous fun stuff on math and astronomy.
Math: if you haven't seen a sphere turn inside out, you've got
to watch this classic movie, now available for free online:
1) The Geometry Center, Outside in,
http://video.google.com/videoplay?docid=6626464599825291409
Astronomy: did you ever wonder where dust comes from? I'm
not talking about dust bunnies under your bed  I'm talking
about the dust cluttering our galaxy, which eventually clumps
together to form planets and... you and me!
These days most dust comes from aging stars called "asymptotic giant
branch" stars. The sun will eventually become one of these. The
story goes like this: first it'll keep burning until the hydrogen in
its core is exhausted. Then it'll cool and become a red giant.
Eventually helium at the core will ignite, and the Sun will heat up
and shrink again... but its core will then become cluttered with even
heavier elements, so it'll cool and expand once more, moving onto the
"asymptotic giant branch". At this point it'll have a layered
structure: heavier elements near the bottom, then a layer of helium,
then hydrogen on the top.
(A similar fate awaits any star between 0.6 and 10 solar masses,
though the details depend on the mass. For the more dramatic
fate of heavier stars, see "week204".)
Anyway: this layered structure is unstable, so asymptotic giant
branch stars pulse every 10 to 100 thousand years or so. And, they
puff out dust! Stellar wind then blows this dust out into space.
A great example is the Red Rectangle:
2) Rungs of the Red Rectangle, Astronomy picture of the day,
May 13, 2004, http://apod.nasa.gov/apod/ap040513.html
Here two stars 2300 light years from us are spinning around
each other while pumping out a huge torus of icy dust grains and
hydrocarbon molecules. It's not really shaped like a rectangle
or X  it just looks that way. The scene is about 1/3 of a light
year across.
Ciska MarkwickKemper is an expert on dust. She's an astrophysicist
at the University of Manchester. Together with some coauthors, she
wrote a paper about the Red Rectangle:
3) F. MarkwickKemper, J. D. Green, E. Peeters, Spitzer
detections of new dust components in the outflow of the Red
Rectangle, Astrophys. J. 628 (2005) L119L122. Also available
as arXiv:astroph/0506473.
They used the Spitzer Space Telescope  an infrared telescope on
a satellite in earth orbit  to find evidence of magnesium and
iron oxides in this dust cloud.
But, what made dust in the early Universe? It took about a
billion years after the Big Bang for asymptotic giant branch stars
to form. But we know there was a lot of dust even before then!
We can see it in distant galaxies lit up by enormous black holes
called "quasars", which pump out vast amounts of radiation as
stuff falls into them.
MarkwickKemper and coauthors have also tackled that question:
4) F. MarkwickKemper, S. C. Gallagher, D. C. Hines and J. Bouwman,
Dust in the wind: crystalline silicates, corundum and periclase in
PG 2112+059, Astrophys. J. 668 (2007), L107L110. Also available
as arXiv:0710.2225.
They used spectroscopy to identify various kinds of dust in
a distant galaxy: a magnesium silicate that geologists call
"forsterite", a magnesium oxide called "periclase", and aluminum
oxide, otherwise known as "corundum"  you may have seen it on
sandpaper.
And, they hypothesize that these dust grains were formed in the
hot wind emanating from the quasar at this galaxy's core!
So, besides being made of star dust, as in the Joni Mitchell
song, you also may contain a bit of black hole dust.
Okay  now that we've got that settled, on to London!
Minhyong Kim is a friend I met back in 1986 when he was a grad
student at Yale. After dabbling in conformal field theory, he
became a student of Serge Lang and went into number theory. He
recently moved to England and started teaching at University
College, London. I met him there this summer, in front of the
philosopher Jeremy Bentham, who had himself mummified and stuck
in a wooden cabinet near the school's entrance.
If you're not into number theory, maybe you should read this:
5) Minhyong Kim, Why everyone should know number theory,
available at http://www.ucl.ac.uk/~ucahmki/numbers.pdf
Personally I never liked the subject until I realized it was
a form of *geometry*. For example, when we take an equation like
this
x^2 + y^3 = 1
and look at the real solutions, we get a curve in the plane 
a "real curve". If we look at the complex solutions, we get
something bigger. People call it a "complex curve", because
it's analogous to a real curve. But topologically, it's
2dimensional. This will be important in a few minutes, so
don't forget it!
If we use polynomial equations with more variables, we get
higherdimensional shapes called "algebraic varieties"  either
real or complex. Either way, we can study these shapes using
geometry and topology.
But in number theory, we might study the solutions of these
equations in some other number system  for example in Z/p,
meaning the integers modulo some prime p. At first glance there's
no geometry involved anymore. After all, there's just a *finite
set* of solutions! However, algebraic geometers have figured
out how to apply ideas from geometry and topology, mimicking
tricks that work for the real and complex numbers.
All this is very fun and mindblowing  especially when we reach
Grothendieck's idea of "etale topology", developed around 1958.
This is a way of studying "holes" in things like algebraic
varieties over finite fields. Amazingly, it gives results that
nicely match the results we get for the corresponding complex
algebraic varieties! That's part of what the "Weil conjectures"
say.
You can learn the details here:
6) J. S. Milne, Lectures on Etale Cohomology, available at
http://www.jmilne.org/math/CourseNotes/math732.html
Anyway, I quizzed about Minhyong about one of the big mysteries
that's been puzzling me lately. I want to know why the integers
resemble a 3dimensional space  and how prime numbers are like
"knots" in this space!
I made a small step toward explaining this back in "week205".
There I sketched one of the basic ideas of algebraic geometry:
every commutative ring, for example the integers or the integers
modulo p, has a kind of space associated to it, called its "spectrum".
We can think of elements of the commutative ring as functions on
this space. I also explained why the process turning a commutative
ring into a space is "contravariant". This implies that the obvious
map from the integers to the integers modulo p
Z > Z/p
gives rise to a map going the other way between spectra:
Spec(Z/p) > Spec(Z)
In "week218" I reviewed an old argument saying that Spec(Z) is
analogous to the complex plane, and that Spec(Z/p) is analogous to a
point. From this viewpoint, primes gives something like points in a
plane.
However, from a different viewpoint, primes give something like
circles in a 3d space!
The easy thing to see is how Spec(Z/p) acts more like a circle than a
point. In particular, its "etale topology" resembles the topology of a
circle. Oversimplifying a bit, the reason is that just as the circle
has one nfold cover for each integer n > 0, so too does Spec(Z/p).
To get the nfold cover of the circle, you just wrap it around itself
n times. To get the nfold cover of Spec(Z/p), we take the spectrum of
the field with p^n elements, which is called F_{p^n}. Z/p sits inside
this larger field:
Z/p > F_{p^n}
so by the contravariance I mentioned, we get a map going the other
way:
Spec(F_{p^n}) > Spec(Z/p)
which is our nfold cover.
I should explain this in much more detail someday  it involves the
relation between etale cohomology, Galois theory and covering
spaces. I began tackling this in "week213", but I have a long way
to go.
Anyway, the basic idea here is that each prime p gives a "circle"
Spec(Z/p) sitting inside Spec(Z). But the really nonobvious part
is that according to etale cohomology, Spec(Z) is *3dimensional* 
and the different circles corresponding to different primes are
*linked*!
I've been fascinated by this ever since I heard about it, but I
got even more interested when I saw a draft of a paper by
Kapranov and Smirnov. I got it from Thomas Riepe, who got
it from Yuri Manin. There's a version on the web:
7) M. Kapranov and A. Smirnov, Cohomology determinants and
reciprocity laws: number field case, available at
http://wwwhomes.unibielefeld.de/triepe/F1.html
It begins:
The analogies between number fields and function fields
have been a longtime source of inspiration in arithmetic.
However, one of the most intriguing problems in this
approach, namely the problem of the absolute point, is
still far from being satisfactorialy understood. The
scheme Spec(Z), the final object in the category of schemes,
has dimension 1 with respect to the Zariski topology
and at least 3 with respect to the etale topology. This
has generated a longstanding desire to introduce a more
mythical object P, the "absolute point", with a natural
morphism X > P given for any arithmetic scheme X [...]
Even though I don't fully understand this, I can tell something
big is afoot here. Spec(Z) is the "space"  really a "scheme" 
corresponding to the integers. And, I think they're saying that
because Spec(Z) is so big and fancy from the viewpoint of
etale topology, there should be some mysterious kind of "point"
that's much smaller than Spec(Z)  the "absolute point".
Anyway, in this paper the authors explain how the "Legendre
symbol" of primes is analogous to the "linking number" of knots.
The Legendre symbol depends on two primes: it's 1 or 1 depending
on whether or not the first is a square modulo the second. The
linking number depends on two knots: it says how many times the
first winds around the second.
The linking number stays the same when you switch the two knots.
The Legendre symbol has a subtler symmetry when you switch the
two primes: this symmetry is called "quadratic reciprocity", and
it has lots of proofs, starting with a bunch by Gauss  all a bit
tricky.
I'd feel very happy if I truly understood why quadratic reciprocity
reduces to the symmetry of the linking number when we think of
primes as analogous to knots. Unfortunately, I'll need to think a
lot more before I really get the idea. I got into number theory
late in life, so I'm pretty slow at it.
This paper studies subtler ways in which primes can be "linked":
8) Masanori Morishita, Milnor invariants and Massey products for
prime numbers, Compositio Mathematica 140 (2004), 6983.
You may know the Borromean rings, a design where no two rings are
linked in isolation, but all three are when taken together. Here
the linking numbers are zero, but the linking can be detected by
something called the "Massey triple product". Morishita
generalizes this to primes!
But I want to understand the basics...
The secret 3dimensional nature of the integers and certain other
"rings of algebraic integers" seems to go back at least to the work
of Artin and Verdier:
9) Michael Artin and JeanLouis Verdier, Seminar on etale cohomology
of number fields, Woods Hole, 1964.
You can see it clearly here, starting in section 2:
10) Barry Mazur, Notes on the etale cohomology of number fields,
Annales Scientifiques de l'Ecole Normale Superieure Ser. 4,
6 (1973), 521552. Also available at
http://www.numdam.org/numdambin/fitem?id=ASENS_1973_4_6_4_521_0
By now, a big "dictionary" relating knots to primes has been
developed by Kapranov, Mazur, Morishita, and Reznikov. This
seems like a readable introduction:
11) Adam S. Sikora, Analogies between group actions on 3manifolds
and number fields, available as arXiv:math/0107210.
I need to study it. These might also be good  I haven't looked
at them yet:
12) Masanori Morishita, On certain analogies between knots and
primes, J. Reine Angew. Math. 550 (2002), 141167.
Masanori Morishita, On analogies between knots and primes,
Sugaku 58 (2006), 4063.
After giving a talk on 2Hilbert spaces at University College, I went
to dinner with Minhyong and some folks including Ray Streater. Ray
Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and
All That". Like almost every mathematician who has seriously tried to
understand quantum field theory, I've learned a lot from this book.
So, it was fun meeting Streater, talking with him  and finding out
he'd once been made an honorary colonel of the US Army to get a free
plane trip to the Rochester Conference! This was a big important
particle physics conference, back in the good old days.
He also described Geoffrey Chew's Rochester conference talk on the
analytic Smatrix, given at the height of the bootstrap model fad.
Wightman asked Chew: why assume from the start that the Smatrix was
analytic? Why not try to derive it from simpler principles? Chew
replied that "everything in physics is smooth". Wightman asked about
smooth functions that aren't analytic. Chew thought a moment and
replied that there weren't any.
Hahaha...
What's the joke? Well, first of all, Wightman had already succeeded
in deriving the analyticity of the Smatrix from simpler principles.
Second, any good mathematician  but not necessarily every physicist,
like Chew  will know examples of smooth functions that aren't
analytic.
Anyway, Streater has just finished an interesting book on "lost
causes" in physics: ideas that sounded good, but never panned out.
Of course it's hard to know when a cause is truly lost. But a
good pragmatic definition of a lost cause in physics is a topic
that shouldn't be given as a thesis problem.
So, if you're a physics grad student and some professor wants you to
work on hidden variable theories, or octonionic quantum mechanics,
or deriving laws of physics from Fisher information, you'd better
read this:
13) Ray F. Streater, Lost Causes in and Beyond Physics, Springer
Verlag, Berlin, 2007.
(I like octonions  but I agree with Streater about not inflicting
them on physics grad students! Even though all my students are in
the math department, I still wouldn't want them working mainly on
something like that. There's a lot of more general, clearly useful
stuff that students should learn.)
I also spoke to Andreas Doering and Chris Isham about their work
on topos theory and quantum physics. Andreas Doering lives near
Greenwich, while Isham lives across the Thames in London proper.
So, I talked to Doering a couple times, and once we visited Isham
at his house.
I mainly mention this because Isham is one of the gurus of quantum
gravity, profoundly interested in philosophy... so I was surprised,
at the end of our talk, when he showed me into a room with a huge
rack of computers hooked up to a bank of about 8 video monitors,
and controls reminiscent of an airplane cockpit.
It turned out to be his homemade flight simulator! He's been a
hobbyist electrical engineer for years  the kind of guy who
loves nothing more than a soldering iron in his hand. He'd just
gotten a big 750watt power supply, since he'd blown out his
previous one.
Anyway, he and Doering have just come out with a series of papers:
14) Andreas Doering and Christopher Isham, A topos foundation
for theories of physics: I. Formal languages for physics,
available as arXiv:quantph/0703060.
II. Daseinisation and the liberation of quantum theory,
available as arXiv:quantph/0703062.
III. The representation of physical quantities with arrows,
available as arXiv:quantph/0703064.
IV. Categories of systems, available as arXiv:quantph/0703066.
Though they probably don't think of it this way, you can think
of their work as making precise Bohr's ideas on seeing the quantum
world through classical eyes. Instead of talking about all
observables at once, they consider collections of observables that
you can measure simultaneously without the uncertainty principle
kicking in. These collections are called "commutative subalgebras".
You can think of a commutative subalgebra as a classical snapshot
of the full quantum reality. Each snapshot only shows part of the
reality. One might show an electron's position; another might show
its momentum.
Some commutative subalgebras contain others, just like some open
sets of a topological space contain others. The analogy is a good
one, except there's no one commutative subalgebra that contains
*all* the others.
Topos theory is a kind of "local" version of logic, but where the
concept of locality goes way beyond the ordinary notion from
topology. In topology, we say a property makes sense "locally"
if it makes sense for points in some particular open set.
In the DoeringIsham setup, a property makes sense "locally" if
it makes sense "within a particular classical snapshot of reality" 
that is, relative to a particular commutative subalgebra.
(Speaking of topology and its generalizations, this work on topoi and
physics is related to the "etale topology" idea I mentioned a while
back  but technically it's much simpler. The etale topology lets
you define a topos of "sheaves" on a certain category. The
DoeringIsham work just uses the topos of "presheaves" on the poset
of commutative subalgebras. Trust me  while this may sound scary,
it's much easier.)
Doering and Isham set up a whole program for doing physics
"within a topos", based on existing ideas on how to do math in
a topos. You can do vast amounts of math inside any topos just
as if you were in the ordinary world of set theory  but using
intuitionistic logic instead of classical logic. Intuitionistic
logic denies the principle of excluded middle, namely:
"For any statement P, either P is true or not(P) is true."
In Doering and Isham's setup, if you pick a commutative subalgebra
that contains the position of an electron as one of its observables,
it can't contain the electron's momentum. That's because these
observables don't commute: you can't measure them both simultaneously.
So, working "locally"  that is, relative to this particular
subalgebra  the statement
P = "the momentum of the electron is zero"
is neither true nor false! It's just not defined.
Their work has inspired this very nice paper:
15) Chris Heunen and Bas Spitters, A topos for algebraic quantum
theory, available as arXiv:0709.4364.
so let me explain that too.
I said you can do a lot of math inside a topos. In particular,
you can define an algebra of observables  or technically, a
"C*algebra".
By the IshamDoering work I just sketched, any C*algebra of
observables gives a topos. Heunen and Spitters show that the
original C*algebra gives a C*algebra in this topos, which
is commutative even if the original one was noncommutative!
That actually makes sense, since in this setup each "local view"
of the full quantum reality is classical.
What's really neat is that the GelfandNaimark theorem, saying
commutative C*algebras are always algebras of continuous functions
on compact Hausdorff spaces, can be generalized to work within
any topos. So, we get a space *in our topos* such that observables
of the C*algebra *in the topos* are just functions on this space.
I know this sounds technical if you're not into this stuff. But
it's really quite wonderful. It basically means this: using topos
logic, we can talk about a classical space of states for a quantum
system! However, this space typically has "no global points" 
that's called the "KochenSpecker theorem". In other words,
there's no overall classical reality that matches all the classical
snapshots.
As you can probably tell, category theory is gradually seeping
into this post, though I've been doing my best to keep it
hidden. Now I want to say what Eugenia Cheng explained on
that train to Sheffield. But at this point, I'll break down and
assume you know some category theory  for example, monads.
If you don't know about monads, never fear! I defined them in
"week89", and studied them using string diagrams in "week92".
Even better, Eugenia Cheng and Simon Willerton have formed a
little group called the Catsters  and under this name, they've
put some videos about monads and string diagrams onto YouTube!
This is a really great new use of technology. So, you should
also watch these:
16) The Catsters, Monads,
http://youtube.com/view_play_list?p=0E91279846EC843E
The Catsters, Adjunctions,
http://youtube.com/view_play_list?p=54B49729E5102248
The Catsters, String diagrams, monads and adjunctions,
http://youtube.com/view_play_list?p=50ABC4792BD0A086
A very famous monad is the "free abelian group" monad
F: Set > Set
which eats any set X and spits out the free abelian group on X,
say F(X). A guy in F(X) is just a formal linear combination
of guys in X, with integer coefficients.
Another famous monad is the "free monoid" monad
G: Set > Set
This eats any set X and spits out the free monoid on X, namely
G(X). A guy in G(X) is just a formal product of guys in X.
Now, there's yet another famous monad, called the "free
ring" monad, which eats any set X and spits out the free ring on
this set. But, it's easy to see that this is just F(G(X))!
After all, F(G(X)) consists of formal linear combinations of
formal products of guys in X. But that's precisely what you find
in the free ring on X.
But why is FG a monad? There's more to a monad than just a
functor. A monad is really a kind of *monoid* in the world of
functors from our category (here Set) to itself. In particular,
since F is a monad, it comes with a natural transformation called
a "multiplication":
m: FF => F
which sends formal linear combinations of formal linear combinations
to formal linear combinations, in the obvious way. Similarly,
since G is a monad, it comes with a natural transformation
n: GG => G
sending formal products of formal products to formal products.
But how does FG get to be a monad? For this, we need some
natural transformation from FGFG to FG!
There's an obvious thing to try, namely
mn
FGFG ======> FFGG ======> FG
where in the first step we switch G and F somehow, and in the
second step we use m and n. But, how do we do the first step?
We need a natural transformation
d: GF => FG
which sends formal products of formal linear combinations
to formal linear combinations of formal products. Such a
thing obviously exists; for example, it sends
(x + 2y)(x  3z)
to
xx + 2yx  3xz  6yz
It's just the distributive law!
Quite generally, to make the composite of monads F and G
into a new monad FG, we need something that people call a
"distributive law", which is a natural transformation
d: GF => FG
This must satisfy some equations  but you can work out
those yourself. For example, you can demand that
FdG mn
FGFG ======> FFGG ======> FG
make FG into a monad, and see what that requires. (Besides the
"multiplication" in our monad, we also need the "unit", so you
should also think about that  I'm ignoring it here because it's
less sexy than the multiplication, but it's equally essential.)
However: all this becomes more fun with string diagrams!
As the Catsters explain, and I explained in "week89", the
multiplication m: FF => F can be drawn like this:
\ /
\ /
F\ F/
\ /
\ /
\ /
\ /
\ /
m





F

And, it has to satisfy the associative law, which says we
get the same answer either way when we multiply three things:
\ / / \ \ /
\ / / \ \ /
F\ /F F/ F\ F\ /F
\/ / \ \/
m\ / \ /m
\ / \ /
F\ / \ /F
\ / \ /
m m
 
 = 
 
 
 
F F
 
The multiplication n: GG => G looks similar to m, and it too has
to satisfy the associative law.
How do we draw the distributive law d: FG => GF? Since it's a
process of switching two things, we draw it as a *braiding*:
F\ /G
\ /
/
/ \
G/ \F
I hope you see how incredibly cool this is: the good old
distributive law is now a *braiding*, which pushes our diagrams
into the third dimension!
Given this, let's draw the multiplication for our wouldbe
monad FG, namely
FdG mn
FGFG ======> FFGG ======> FG
It looks like this:
\ \ / /
\ \ / /
F\ G\ F/ /G
\ \ / /
\ \ / /
\ \ / /
\ / /
\ / \ /
m n
 
 
 
 
 
F G
 
Now, we want *this* multiplication to be associative! So,
we need to draw an equation like this:
\ / / \ \ /
\ / / \ \ /
\ / / \ \ /
\/ / \ \/
\ / \ /
\ / \ /
\ / \ /
\ / \ /
 
 
 = 
 
 
 
 
 
but with the strands *doubled*, as above  I'm too lazy to draw
this here. And then we need to find some nice conditions that
make this associative law true. Clearly we should use the
associative laws for m and n, but the "braiding"  the
distributive law d: FG => GF  also gets into the act.
I'll leave this as a pleasant exercise in string diagram
manipulation. If you get stuck, you can peek in the back of
the book:
17) Wikipedia, Distibutive law between monads,
http://en.wikipedia.org/wiki/Distributive_law_between_monads
The two scary commutative rectangles on this page are the
"nice conditions" you need. They look nicer as string
diagrams. One looks like this:
F\ G\ /G F\ G/ /G
\ \ / \ / /
\ n \ / /
\ / / /
\ / = / \ /
/ / /
/ \ / /
/ \ \ / \
/ \ \ / \
G/ \F n \F
/ \ G \
In words:
"multiply two G's and slide the result over an F" =
"slide both the G's over the F and then multiply them"
If the pictures were made of actual string, this would be obvious!
The other condition is very similar. I'm too lazy to draw it,
but it says
"multiply two F's and slide the result under a G" =
"slide both the F's under a G and then multiply them"
All this is very nice, and it goes back to a paper by Beck:
18) Jon Beck, Distributive laws, Lecture Notes in Mathematics
80, Springer, Berlin, 1969, pp. 119140.
This isn't what Eugenia explained to me, though  I already knew
this stuff. She started out by explaining something in a paper
by Street:
19) Ross Street, The formal theory of monads, J. Pure Appl. Alg.
2 (1972), 149168.
which is reviewed at the beginning here:
20) Steve Lack and Ross Street, The formal theory of monads II,
J. Pure Appl. Alg. 175 (2002), 243265. Also available at
http://www.maths.usyd.edu.au/u/stevel/papers/ftm2.html
(Check out the cool string diagrams near the end!)
Street noted that for any category C, there's a category Mnd(C)
whose objects are monads on C and whose morphisms are "monad
transforms": functors from C to C that make an obvious square
commute.
And, he noted that a monad on Mnd(C) is a pair of monads on C
related by a distributive law!
That's already mindbogglingly beautiful. According to Eugenia,
it's in the last sentence of Street's paper. But in her new work:
21) Eugenia Cheng, Iterated distributive laws, available as
arXiv:0710.1120.
she goes a bit further: she considers monads in Mnd(Mnd(C)),
and so on. Here's the punchline, at least for today: she shows
that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related
by distributive laws satisfying the YangBaxter equation:
\F G/ H F G\ /H
\ /   \ /
/   /
/ \   / \
/ \  \ / \
 \ / \ / 
 / = / 
 / \ / \ 
 / \ / \ 
\ /   \ /
\ /   \ /
/   /
/ \   / \
/H \G F H G/ \F
This is also just what you need to make the composite FGH
into a monad!
By the way, the pathetic piece of ASCII art above is lifted
from "week1", where I first explained the YangBaxter equation.
That was back in 1993. So, it's only taken me 14 years to learn
that you can derive this equation from considering monads on
the category of monads on the category of monads on a category.
You may wonder if this counts as progress  but Eugenia
studies lots of *examples* of this sort of thing, so it's far
from pointless.
Okay... finally, the Tale of Groupoidification. I'm a bit tired
now, so instead of telling you more of the tale, let me just say
the big news.
Starting this fall, James Dolan and I are running a seminar on
geometric representation theory, which will discuss:
Actions and representations of groups, especially symmetric groups
Hecke algebras and Hecke operators
Young diagrams
Schubert cells for flag varieties
qdeformation
Spans of groupoids and groupoidification
This is the Tale of Groupoidification in another guise.
Moreover, the Catsters have inspired me to make videos of this
seminar! You can already find some here, along with course
notes and blog entries where you can ask questions and talk about
the material:
22) John Baez and James Dolan, Geometric representation theory seminar,
http://math.ucr.edu/home/baez/qgfall2007/
More will show up in due course. I hope you join the fun.

Quote of the Week:
It is a glorious feeling to discover the unity of a set of phenomena
that at first seem completely separate.  Albert Einstein

Addenda: You can find some of Streater's "lost causes in
physics" online:
23) Ray F. Streater, Various causes in physics and elsewhere,
http://www.mth.kcl.ac.uk/~streater/causes.html
For the proof of the GelfandNaimark theorem inside a topos, see:
24) Bernhard Banachewski and Christopher J. Mulvey, A globalisation
of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006),
62103. Also available at
http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf
They show that any commutative C*algebra A in a Grothendieck topos is
canonically isomorphic to the C*algebra of continuous complex functions
on the compact, completely regular locale that is its maximal spectrum
(that is, the space of homomorphisms f: A > C). Conversely, they show
any compact completely regular locale X gives a commutative C*algebra
consisting of continuous complex functions on X. Even better, they
explain what all this stuff means.
Jordan Ellenberg sent me the following comments about knots and primes:
1. In the viewpoint of Deninger, very badly oversimplified, Spec Z
is to be thought of not just as a 3manifold but as a 3manifold with
a flow, in which the primes are not just knots, but are precisely the
closed orbits of the flow!
I may have vaguely heard of that... I'll have to look into it.
Thanks!
2. One thing to keep in mind about the analogy is that "the
complement of a knot or link in a 3manifold" and "the complement of
a prime or composite integer in Spec Z" (which is to say Spec Z[1/N])
are both "things which have fundamental groups," thanks to
Grothendieck in the latter case. And much of the concrete part of
the analogy (like the stuff about linking numbers) follows from this
fact.
3. On a similar note, a recent paper of Dunfield and Thurston which
I like a lot, "Finite covers of random 3manifolds," develops a model
of "random 3manifold" and shows that the behavior of the first
homology of a random 3manifold mod p is exactly the same as the
_predicted_ behavior of the mod p class group of a random number
field under the CohenLenstra heuristics. In other words, you should
not think of Spec Z or Spec Z[1/N] as being anything like a
_particular_ 3manifold  better to think of the class of 3
manifolds as being like the class of number fields.
Here's one of Deninger's papers:
25) Christopher Deninger, Number theory and dynamical systems on
foliated spaces, available as arXiv:math/0204110.
And here's the paper by Dunfield and Thurston:
26) Nathan M. Dunfield and William P. Thurston, Finite covers of
random 3manifolds, available as arXiv:math/0502567.
On the nCategory Cafe, a number theorist named James caught
some serious mistakes in the original version of this Week's
Finds. I've corrected those, mainly by leaving out a screwedup
attempt to explain why Spec(Z) is 3dimensional. Here are his
remarks on this mystery:
So then why should there be the two dimensions of primes needed
to make Spec(Z) threedimensional? I don't think there is a
purethought answer to this question. As you wrote, there is a
scientific answer in terms of ArtinVerdier duality, which is
pretty much the same as class field theory. There is also a
purethought answer to an analogous question. Let me try to
explain that.
Instead of considering Z, let's consider F[x], where F is a finite
field. They are both principal ideal domains with finite residue
fields, and this makes them behave very similarly, even on a deep
level. I'll explain why F[x] is threedimensional, and then by
analogy we can hope Z is, too. Now F[x] is an Falgebra. In
other words, X = Spec(F[x]) is a space mapping to S = Spec(F).
I already explained why S is a circle from the point of view
of the etale topology. So, if X is supposed to be threedimensional,
the fibers of this map better be twodimensional. What are the
fibers of this map? Well, what are the points of S? A point
in the etale topology is Spec of some field with a trivial absolute
Galois group, or in other words, an algebraically closed field
(even better, a separably closed one). Therefore a etale point
of S is the same thing as Spec of an algebraic closure Fbar of F.
What then is the fiber of X over this point? It's Spec of the
ring Fbar[x]. Now, *this* is just the affine line over an
algebraically closed field, so we can figure out its cohomological
dimension. The affine line over the complex numbers, another
algebraically closed field, is a plane and therefore has cohomological
dimension 2. Since etale cohomology is kind of the same as usual
singular cohomology, the etale cohomological dimension of
Spec(Fbar[x]) ought to be 2.
Therefore X looks like a 3manifold fibered in 2manifolds over
Spec(F), which looks like a circle. Back to Spec(Z), we
analogously expect it to look like a 3manifold, but absent a
(nonformal) theory of the field with one element, Z is not
an algebra over anything. Therefore we expect Spec(Z) to
be a 3manifold, but not fibered over anything.
For more discussion, go to the nCategory Cafe:
http://golem.ph.utexas.edu/category/2007/10/this_weeks_finds_in_mathematic_18.html

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html