Also available as http://math.ucr.edu/home/baez/week260.html
December 24, 2007
This Week's Finds in Mathematical Physics (Week 260)
John Baez
Since it's Christmas Eve, I thought I'd list some free books you
can download. I'm a big fan of giving the world presents... and
I'm not the only one.
But first, this week's nebulae! Here's one called the Retina:
1) Retina Nebula, Hubble Heritage Project,
http://heritage.stsci.edu/2002/14/
This is actually a tube of ionized gas about a quarter of a lightyear
across and one lightyear long. It's a planetary nebula produced
by a dying star. If you zoom in and look closely, you can see this
star lurking in the middle, now a mere white dwarf.
The blue light is the most energetic, so it's really hot where you see
blue. This blue light comes from singly ionized helium  helium where
one electron has been knocked off. The green light is a bit less
energetic: that's from doubly ionized oxygen. The red light comes from
even cooler regions: that's from singly ionized nitrogen.
You can also see a lot of "dust lanes" in this photo. They're beautiful.
And they're big! The width of each one is about 160 times the distance
between the Sun and the Earth. The gas and dust in these lanes is about
1000 times higher than elsewhere. But what creates them?
Apparently, when the fastmoving glowing hot gas from the star crashes
into the invisible gas in the surrounding interstellar space, the
boundary gets sort of crumpled, and these dust lanes form. It's vaguely
similar to the puffy surface of a cumulus cloud. But here the mechanism
is different, because it involves a "shock wave": the hot gas is moving
faster than the speed of sound as it hits the cold gas!
This effect is called a "Vishniac instability", since in 1983, the
astrophysicist Ethan Vishniac showed that a shock wave moving in a
sufficiently compressible medium would be subject to an instability
of this sort, growing as the square root of time. I've never seen
how Vishniac's calculations work, so the mathematics underlying this
beautiful phenomenon will have to wait for another day.
Note that this planetary nebula, like the others I've shown you, is
far from spherically symmetric. Astrophysicists used to pretend stars
were spherically symmetric. But, that's a bad approximation whenever
anything really exciting happens... just like in the old joke where
the punchline is "consider a spherical cow".
As I said, the Retina Nebula is actually shaped like a tube. Viewed
from either end, this tube would look very different  probably like
the Ring Nebula:
2) Ring Nebula, Hubble Heritage Project,
http://heritage.stsci.edu/1999/01/
This is one lightyear across. Again we see He II blue light with a
wavelength of 4686 angstroms, then O III green light at 5007
angstroms, then N II red light at 6584 angstroms. You can also see
the white dwarf as a tiny dot in the center; it's about 100,000 kelvin
in temperature.
(In case you're wondering, an "angstrom" is an obsolete but popular
unit of distance, equal to 10^{10} meters. Just like the "parsec",
it's a sign that astronomy is an old science. Anders Jonas Angstrom
was one of the founders of spectroscopy, back around 1860. Archaic
conventions may also explain why single ionized helium is called "He
II", and so on. Maybe the number zero hadn't fully caught on.)
Next: free books!
At least around here, Christmas seems to be all about buying stuff and
giving it away. Giving is good. But I think gifts have more soul if
you make them yourself. This is one of the great things about the
internet: it lets us create things and give them to *everyone in the
world*  or more precisely: everybody who wants them, and nobody who
doesn't.
In this spirit, here's a roundup of free books on math and physics:
gifts from their authors to you. There are lots out there. I'll
only list a few. For more, try these sites:
3) George Cain, Online Mathematics Textbooks,
http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html
4) Free Online Mathematics Books,
http://www.pspxworld.com/book/mathematics/
5) Alex Stefanov, Textbooks in Mathematics,
http://users.ictp.it/~stefanov/mylist.html or (with annoying ads,
but more permanent) http://us.geocities.com/alex_stef/mylist.html
Despite its title, Stefanov's excellent site includes a lot of
books on physics. I can't find lists *specifically* devoted to
free physics books, but there are a lot out there  including a lot on
the arXiv.
Anyway, let's dive in!
What if you're dying to learn physics, but don't know where to start?
Start here:
6) Physics Books Online, http://www.sciencebooksonline.info/physics.html
You'll find plenty of free online books, starting from the basics
and working up to advanced topics. But to dig deeper into these
mysteries, you'll eventually need to learn a bunch of math. Do you
remember what Victor Weisskopf said when a student asked how much math
a physicist needs to know?
"More."
This can be scary when you're just getting started. What if you don't
know calculus, for example?
Simple: learn calculus! This book is a classic  and it's free:
7) Gilbert Strang, Calculus, WellesleyCambridge Press, Cambridge,
1991. Also available at
http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm<
It really explains things clearly. I may use it the next time I
teach calculus. We professors need to quit making our students
buy expensive textbooks, and switch to free online books! We could
join forces and make wiki textbooks that are a lot better and
more flexible than the budgetbusting, backbreaking mammoths we
currently inflict on our kids. But there are already a lot of good
texts available free online.
Or: what if you know calculus, but you're still swimming through the
undergraduate sea of differential equations, Fourier transforms,
matrices, vectors and tensors? Then this should be really helpful:
8) James Nearing, Mathematical Tools for Physics, available at
http://www.physics.miami.edu/~nearing/mathmethods/
Unlike the usual dry and formal textbook, it reads like a friendly
uncle explaining things in plain English, trying to cut through the red
tape and tell you how to actually think about this stuff.
For example, on page 3 he introduces the hyperbolic trig functions:
Where do hyperbolic functions come from? If you have a mass
in equilibrium, the total force on it is zero. If it's in *stable*
equilibrium then if you push it a little to one side and release
it, the force will push it back to the center. If it is *unstable*
then when it's a bit to one side it will be pushed farther away
from the equilibrium point. In the first case, it will oscillate
about the equilibrium position and the function of time will be
a circular trigonometric function  the common sines or cosines of
time, A cos(wt). If the point is unstable, the motion will be
described by hyperbolic functions of time, sinh(wt) instead of
sin(wt). An ordinary ruler held at one end will swing back and
forth, but if you try to balance it at the other end it will fall
over. That's the difference between cos and cosh.
He goes into more detail later, after introducing the complex numbers.
This book also features some great animations of Taylor series and
Fourier series.
There are free online books at all levels... so let's soar a bit
higher. How about if you're a more advanced student trying to learn
general relativity? Here you go:
9) Sean M. Carroll, Lecture Notes on General Relativity, available as
arXiv:grqc/9712019
How about quantum field theory? Then you're in luck  there are
*two* detailed books available online:
10) Warren Siegel, Fields, available as arXiv:hepth/9912205
10) Mark Srednicki, Quantum Field Theory, Cambridge U. Press,
Cambridge, 2007. Also available at
http://www.physics.ucsb.edu/~mark/qft.html
Or what about algebraic topology? Again you're in luck, since you
can read both Allen Hatcher's gentle introduction and Peter May's
highpowered "concise course":
11) Allen Hatcher, Algebraic Topology, Cambridge U. Press, Cambridge,
2002. Also available at
http://www.math.cornell.edu/~hatcher/AT/ATpage.html
12) Peter May, A Concise Course in Algebraic Topology, U. of Chicago
Press, Chicago, 1999. Also available at
http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
May has a lot of more advanced topology books available at his website,
too  like this classic, where he used operads to solve important
problems involving loop spaces:
13) Peter May, The Geometry of Iterated Loop Spaces, Lecture Notes
in Mathematics 271, Springer, Berlin, 1972. Also available at
http://www.math.uchicago.edu/~may/BOOKS/gils.pdf
Or say you want to learn about vector bundles and how they show up
in physics, from the basics all the way to fancy stuff like Dbranes
and Ktheory? Try this  it's a great sequel to Husemoller's classic
intro to fiber bundles:
14) Dale Husemoller, Michael Joachim, Branislav Jurco and Martin
Schottenloher, Basic Bundle Theory and KCohomology Invariants,
Lecture Notes in Physics 726, Springer, Berlin, 2008. Also
available at
http://www.mathematik.unimuenchen.de/~schotten/Texte/9783540749554_Book_LNP726.pdf
The list goes on and on! The American Mathematical Society will give
you books for free if you prove that you're not a robot by solving a
little puzzle:
15) American Mathematical Society, Books Online By Subject,
http://www.ams.org/online_bks/online_subject.html
Apparently they don't want robots learning advanced math and putting
us professors out of business by teaching with more charisma and
flair. (By the way: make sure to let them put cookies on your
web browser, or they'll send you an endless succession of these
puzzles, without explaining why!)
Since James Dolan and I plan to explain symmetric groups and their
Hecke algebras in our online seminar, this particular book from the
AMS caught my eye:
16) David M. Goldschmidt, Group Characters, Symmetric Functions,
and the Hecke Algebra, AMS, Providence, Rhode Island, 1993.
Also available as http://www.ams.org/online_bks/ulect4/
Since we're also struggling to understand the Langlands program,
this looks good too:
17) Armand Borel, Automorphic Forms, Representations, and Lfunctions,
AMS, 2 volumes, Providence, Rhode Island, 1979. Also available at
http://www.ams.org/online_bks/pspum331/ and
http://www.ams.org/online_bks/pspum332/
It's a serious collection of expository papers by bigshots like
Borel, Cartier, Deligne, Jacquet, Knapp, Langlands, Lusztig, Tate,
Tits, Zuckerman, and many more.
"Motives" are the mysterious virtual building blocks that algebraic
varieties are built from. If you're ready to learn about motives 
I'm not sure I am  try this:
18) Marc Levine, Mixed Motives, AMS, Providence, Rhode Island, 1998.
Also available at http://www.ams.org/online_bks/surv57/
Or, if you're interested in using category theory to make analysis
clearer and more beautiful, try this:
19) Andreas Kriegl and Peter W. Michor, The Convenient Setting of
Global Analysis, AMS, Providence, Rhode Island, 1997. Also available
at http://www.ams.org/online_bks/surv53/
The focus is on getting and working with a "convenient category" of
infinitedimensional manifolds. The idea of a "convenient category"
goes back to topology: at some point, people realized they wanted
this property to hold:
C(X x Y, Z) = C(X, C(Y,Z))
Here C(X,Y) is the space of maps from X to Y. So, the equation above
 really an isomorphism  says that a map from X x Y to Z should
correspond to a map from X to C(Y,Z). A category with this property
is called "cartesian closed". While it may not be obvious why, this
property is so wonderful that people threw out the category of
topological spaces and continuous maps and replaced it with a slightly
different one, just to get this to hold.
Another sort of "convenient category" for differential geometry uses
infinitesimals. Again, you can learn about this in a free book:
20) Anders Kock, Synthetic Differential Geometry, Cambridge U. Press,
Cambridge, 2006. Also available at http://home.imf.au.dk/kock/
This category is not just cartesian closed  it's a topos!
If you don't know what a topos is, never fear  more free books are
coming to your rescue:
21) Robert Goldblatt, Topoi, the Categorial Analysis of Logic,
Dover, 1983. Also available at
http://historical.library.cornell.edu/cgibin/cul.math/docviewer?did=Gold010
22) Michael Barr and Charles Wells, Toposes, Triples and Theories,
Springer, Berlin, 1983. Also available at
http://www.case.edu/artsci/math/wells/pub/ttt.html
The first one is so gentle it makes a good introduction to category
theory as a whole. The second scared the bejeezus out of me for a
decade, but now I like it.
I like Jordan algebras, so I was also pleased to see this classic
offered for free at the AMS website:
23) Nathan Jacobson, Structure and Representations of Jordan Algebras,
AMS, Providence, Rhode Island, 1968. Also available at
http://www.ams.org/online_bks/coll39/
Fans of exceptional Lie algebras will like the last two chapters, on
"connections with Lie algebras" and "exceptional Jordan algebras".
Speaking of Lie algebras, I'd never seen this textbook before:
24) Shlomo Sternberg, Lie Algebras,
http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf
It's a somewhat quirky introduction, not for beginners I think, but
it features some nice special topics: character formulas, the Kostant
Dirac operator, and a detailed study of the center of the universal
enveloping algebra.
This intro to Lie groups is also a bit quirky, but if you like Feynman
diagrams or spin networks, it's irreplaceable:
25) Predrag Cvitanovic, Birdtracks, Lie's, and Exceptional Groups,
available at http://www.nbi.dk/GroupTheory/
One of the great things about this book is that it classifies simple
Lie groups according to their "skein relations"  properties of their
representations, written out diagrammatically. In so doing, Cvitanovic
realized that there's a "magic triangle" containing all the exceptional
Lie groups. This subsumes the "magic square" of Freudenthal and Tits,
which I discussed in "week145" and my octonion webpages.
This idea of Cvitanovic is closely related to the "exceptional series"
of Lie groups  a pattern whose existence was conjectured by Deligne.
This idea of Cvitanovic is closely related to the "exceptional
series" of Lie groups  a pattern whose existence was conjectured
by Deligne. I love the term "exceptional series". It's
an oxymoron, since the exceptional groups were defined as those that
don't fit into any series. But, it makes sense!
To see the exceptional series, it helps to do a mental backflip called
"TannakaKrein duality", where you focus on the category of
representations of the Lie group, instead of the group itself. Then,
draw the morphisms in that category as diagrams, like Feynman
diagrams! Then see what identities they satisfy. New patterns leap
out: new series unify what had been "exceptions".
Very briefly, the idea goes like this. Suppose we have a Lie
group G with Lie algebra L. The Lie bracket takes two elements x and
y and spits out one element [x,y], and it's linear in each variable,
so it gives a linear operator
L tensor L > L
which is actually a morphism in the category of representations of G.
So, following the philosophy of Feynman diagrams, we can draw the
bracket operation like this:
\ /
\ /
\ /



We can even use this to state the definition of a Lie algebra using
diagrams! To say the bracket is antisymmetric:
[y,x] = [x,y]
we just draw this:
\ /  
\ /  
/  
/ \  
/ \ \ /
\ / =  \ /
\ / \ /
 
 
 
To say the Jacobi identity:
[x,[y,z]] = [[x,y],z] + [y,[x,z]]
we just draw this:
\ \ / \ / / \ / /
\ \ / \ / / \ / /
\ \ / \ / / \ /
\ / \ / \ /
\ / = \ / + / \ /
\ / \ / / \ /
  \ /
  \ /
  \ /
  
If that's too cryptic, maybe this will explain what I'm doing:
x y z x y z x y z
\ \ / \ / / \ / /
\ \ / \ / / \ / /
\ \ / \ / / \ /
\ / \ / \ /
\ / = \ / + / \ /
\ / \ / / \ /
  \ /
  \ /
  \ /
  
[x,[y,z]] [[x,y],z] [y,[x,z]]
But in fact, people usually massage this picture to make it even
more cryptic, and call it the "IHX" identity  since the three terms
look like the letters I, H, and X by the time they're done twisting
them around. For a good explanation, with pretty pictures, see:
26) Greg Muller, Chord diagrams and Lie algebras,
http://cornellmath.wordpress.com/2007/12/25/chorddiagramsandliealgebras/
It then turns out that the exceptional Lie algebras F4, E6, E7 and
E8 satisfy yet another identity:
\ /
\ /
\/
 
  =
/\
/ \
/ \
\ / \ /
\ / \ /
\ / 
A  + A  +
/ \ 
/ \ / \
/ \ / \
\ / \ / \ /
\ / \ / \ /
\ / \ / \____/
B / + B   + B ____
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
for various choices of the constants A and B. So, they fit into a
"series"!
I believe the main point of this identity, going back to Vogel's paper
"Algebraic structures on modules of diagrams", is that for these Lie
algebras, the square of the quadratic Casimir is the only degree4
Casimir.
I think there's a lot more to be discovered here, in part by taking
the gnarly computations people have done so far and making them more
beautiful and conceptual. So, I urge all fans of exceptional
mathematics, diagrams, and categories to look at these:
27) Pierre Deligne, La serie exceptionnelle des groupes de Lie,
C. R. Acad. Sci. Paris Ser. I Math 322 (1996), 321326.
Pierre Deligne and R. de Man, The exceptional series of Lie groups II,
C. R. Acad. Sci. Paris Ser. I Math 323 (1996), 577582.
Pierre Deligne and Benedict Gross, On the exceptional series, and its
descendants, C. R. Acad. Sci. Paris Ser. I Math 335 (2002), 877881.
Also available as http://www.math.ias.edu/~phares/deligne/ExcepSeries.ps
29) Pierre Vogel, Algebraic structures on modules of diagrams, 1995.
Available at http://www.institut.math.jussieu.fr/~vogel/ or
http://citeseer.ist.psu.edu/469395.html
The universal Lie algebra, 1999. Available at
http://www.institut.math.jussieu.fr/~vogel/
Vassiliev theory and the universal Lie algebra, 2000.
Available at http://www.institut.math.jussieu.fr/~vogel/
For a good overview, try this:
28) J. M. Landsberg and L. Manivel, Representation theory and projective
geometry, 2002. Available at arXiv:math/0203260.
Alas, they avoid drawing Feynman diagrams, though they talk about them
in section 4. They prefer to use ideas from algebraic geometry:
29) J. M. Landsberg and L. Manivel, The projective geometry of
Freudenthal's magic square, J. Algebra 239 (2001), 477512. Also
available as arXiv:math/9908039.
J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and
Deligne dimension formulas, Adv. Math. 171 (2002), 5985. Also
available as arXiv:math/0107032.
J. M. Landsberg and L. Manivel, Series of Lie groups, available
as arXiv:math/0203241.
Bruce Westbury, whom longtime readers of This Week's Finds will
remember as John Barrett's collaborator, has also worked on this
subject. He has pointed out that both the magic square and the
magic triangle can be given an extra row and column if we introduce
a 6dimensional algebra halfway between the quaternions and the
octonions:
30) Bruce Westbury, Sextonions and the magic square, available
as arXiv:math/0411428.
For even more references, try this:
31) Bruce Westbury, References on series of Lie groups,
http://www.mpimbonn.mpg.de/digitalAssets/2763_references.pdf
This stuff has been on my mind recently, since I've been working on
exceptional groups and grand unified theories with my student
John Huerta. Also, my friend Tevian Dray has a student who just
finished a thesis on a related topic:
32) Aaron Wangberg, The structure of E6, available as arXiv:0711.3447.
In a nutshell: E6 is secretly SL(3,O). Octonions rock!
Happy holidays. Keep learning cool stuff.

Quote of the Week:
If nature has made any one thing less susceptible than all others
of exclusive property, it is the action of the thinking power called
an idea, which an individual may exclusively possess as long as he
keeps it to himself; but the moment it is divulged, it forces itself
into the possession of every one, and the receiver cannot dispossess
himself of it. Its peculiar character, too, is that no one possesses
the less, because every other possesses the whole of it.
Thomas Jefferson

Addenda: Thomas Riepe listed some more free online math books. Tony
Smith pointed out something I already knew, but didn't make clear
above: the idea that E6 is secretly SL(3,O) is far from new.
Thomas wrote:
Some more links:
Milne's great collection (incl. the famous LNM 900), leading the
reader from basic algebra through algebraic number theory, class
fields, modular forms, arithmetic groups,... up to etale cohomology,
Shimura varieties etc:
http://www.jmilne.org/math/index.html
Friedhelm Waldhausen's lectures on algebraic topology and
Ktheory: http://www.math.unibielefeld.de/~fw/
DML: Digital Mathematics Library:
http://www.mathematik.unibielefeld.de/~rehmann/DML/dml_links_author_A.html
G. Harder's math links: http://www.math.unibonn.de/people/harder/
MSRI online books: http://www.msri.org/publications/books/
Finally:
"Nearly three and a half centuries of scientific study and
achievement is now available online in the Royal Society Journals
Digital Archive. This is the longestrunning and arguably most
influential journal archive in Science, including all the back
articles of both Philosophical Transactions and Proceedings":
http://www.pubs.royalsoc.ac.uk/archive
Tony Smith wrote:
Thanks for an interesting list of stuff in week 260,
but I have some questions about
"... 32) Aaron Wangberg, The structure of E6, available as arXiv:
0711.3447.
In a nutshell: E6 is secretly SL(3,O). Octonions rock! ...".
Not only from your brief list descrption, but also from reading the
paper at pages 96 ff
I get the impression that Wangberg is claiming the result E6 = SL(3,O).
Do you get the same impression?
I hope not, and I hope that my impression is somehow mistaken,
because
the result E6 = SL(3,O) is (and has been for some time) well known
and in the literature.
For example, in hepth/9309030 Martin Cederwall and Christian R.
Preitschopf said:
"... It should be possible to realize E6 = SL(3;O) [18,24] on them in
a "spinorlike" manner, much like SO(10) = SL(2;O) acts on its 16
dimensional spinor representations that play the role of homogeneous
coordinates for OP^1 ...
...
18. H. Freudenthal, Adv. Math. 1 (1964) 145.
...
24. A. Sudbery, J. Phys. A17 (1984) 939. ...".
Although that Freudenthal Adv. Math. is listed as a reference in
Wangberg's paper (as reference 5), I did not see the Sudbery paper
listed, and I did not see the Freudenthal reference on page 96.
Please don't misunderstand this message. I think that Wangberg's
thesis is very interesting. I am just trying to get a correct
historical record.
Tony
PS  In Sudbery's 1984 paper, he not only says (at page 950)
"... sl(3,K) ... When K = O, this Lie algebra is a n oncompact form
of the exceptional Lie algebra E6, the maximal compact subalgebra
being F4 ..."
but he goes on to say
"... sp(6,K) ... when K = O it is a noncompact form of E7, the
maximal compact subalgebra being E6 (+) so(2). ...".
For more discussion, go to the nCategory Cafe:
http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_20.html

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