This Week's Finds in Mathematical Physics (Week 260)
- From: baez@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (John Baez)
- Date: Tue, 25 Dec 2007 23:30:02 +0000 (UTC)
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 light-year
across and one light-year 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. But what creates them?
Apparently, when the fast-moving 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 light-year 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) Christoph Schiller, Motion Mountain: The Adventure of Physics,
available free online at http://www.motionmountain.net/
It's an enormous feast of ideas - romantic, wildly ambitious, and
still not finished at 1459 pages. Using a bare minimum of math, it
conveys an enormous amount of physics, all focused on the question
"what is motion?" This question is very deep. We have made
tremendous progress towards answering, but are nowhere near done.
The curious title is explained near the beginning:
The quest to understand motion in all its details and limitations
can be pursued behind a desk, with a book, some paper and a pen.
But to make the adventure more vivid, this text uses the metaphor
of a mountain ascent. Every step towards the top corresponds to
a step towards higher precision in the description of motion. In
addition, with each step the scenery will become more delightful.
At the top of the mountain we shall arrive in a domain where 'space'
and 'time' are words that have lost all meaning and where the sight
of the world's beauty is overwhelming and unforgettable.
Inspiring words. But to dig deeper into such 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, Wellesley-Cambridge 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 budget-busting, back-breaking 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:gr-qc/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:hep-th/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
high-powered "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 D-branes
and K-theory? 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 K-Cohomology Invariants,
Lecture Notes in Physics 726, Springer, Berlin, 2008. Also
available at
http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_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
webbrowswer, 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 L-functions,
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
infinite-dimensional 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's probably not obvious at
first, 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, 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/cgi-bin/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.
Originally this was going to be the topic of this Week's Finds, but
I'm a bit too tired to explain it now. Still, I love the idea. 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
"Tannaka-Krein 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 these diagrams satisfy. New
patterns leap out: new series unify what had been "exceptions".
So, I urge all fans of exceptional mathematics, diagrams, and
categories to look at these:
26) Pierre Deligne, La serie exceptionnelle des groupes de Lie,
C. R. Acad. Sci. Paris Ser. I Math 322 (1996), 321-326.
Pierre Deligne and R. de Man, The exceptional series of Lie groups II,
C. R. Acad. Sci. Paris Ser. I Math 323 (1996), 577-582.
Pierre Deligne and Benedict Gross, On the exceptional series, and its
descendants, C. R. Acad. Sci. Paris Ser. I Math 335 (2002), 877-881.
Also available as http://www.math.ias.edu/~phares/deligne/ExcepSeries.ps
27) 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 of what Landsberg and Manivel have done in this
field, try:
28) J. M. Landsberg and L. Manivel, Representation theory and projective
geometry, 2002. Available at arXiv:math/0203260.
For more details, try these:
29) J. M. Landsberg and L. Manivel, The projective geometry of
Freudenthal's magic square, J. Algebra 239 (2001), 477-512. Also
available as arXiv:math/9908039.
J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and
Deligne dimension formulas, Adv. Math. 171 (2002), 59-85. Also
available as arXiv:math/0107032.
J. M. Landsberg and L. Manivel, Series of Lie groups, available
as arXiv:math/0203241.
For even more, try this:
30) Bruce Westbury, References on series of Lie groups,
http://www.mpim-bonn.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:
31) 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
-----------------------------------------------------------------------
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