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Also available at http://math.ucr.edu/home/baez/week265.html
May 25, 2008
This Week's Finds in Mathematical Physics (Week 265)
John Baez
Today I'd like to talk about the Pythagorean pentagram, Bill Schmitt's
work on Hopf algebras in combinatorics, the magnum opus of Aguiar and
Mahajan, and quaternionic analysis. But first, the astronomy picture
of the week!
I seem to be into moons these days: first Saturn's moon Titan in "week263",
and then Mars' moon Phobos in "week264". On the cosmic scale, our Solar
System is like our back yard. It may not be important in the grand
scheme of things, but we should get to know it and learn to take care of
it. It's got lots of cool moons. So this week, let's talk about Europa:
1) Astronomy Picture of the Day, Gibbous Europa,
http://antwrp.gsfc.nasa.gov/apod/ap071202.html
Europa is the fourth biggest moon of Jupiter, the smallest of the four
seen by Galileo. It's 3000 kilometers in diameter, slightly smaller
than our moon, and it orbits Jupiter once every 3.5 of our days,
though it's almost twice as far from Jupiter as our moon is from us.
It looks like a cracked ball of ice, and that's what it is - at
least near the surface. Indeed, this ancient impact crater looks
like a smashed windshield, or a frozen lake that's been hit with a
sledgehammer:
2) NASA Planetary Photojournal, Ancient impact basin on Europa,
http://photojournal.jpl.nasa.gov/catalog/PIA00702
But this crater, called Tyre, is huge: about as big as the island
of Hawaii, 145 kilometers across! (Beware: this picture is a composite
of three photos taken by the Galileo spacecraft in 1997. It's in false
color designed to show off various structures: the original crater,
the later red cracks, and the blue-green ridges.)
The big question is whether there's liquid water beneath the icy
surface... and if so, maybe life? One model of this moon posits a
solid ice crust. Another says there's liquid water too:
3) NASA Planetary Photojournal, Model of Europa's subsurface structure,
http://photojournal.jpl.nasa.gov/catalog/PIA01669
How can we tell? Europa is the *smoothest* of all solid planets and moons,
with lots of cracks and ridges but few remaining craters. This suggests
either an ocean beneath the surface, or at least ice warm enough to keep
convection going. The region called Conamara Chaos looks like pack ice
here on Earth, hinting at liquid water beneath:
4) NASA Planetary Photojournal, Europa: ice rafting view,
http://photojournal.jpl.nasa.gov/catalog/PIA01127
The bluish white areas have been blanketed with ice dust ejected from
far away when an impact formed a crater called Pwyll. The reddish brown
regions could contain salts or sulfuric acid - it's hard to find out
using spectroscopy, since there's too much ice.
Another very nice piece of evidence for *salty* liquid water inside
Europa is that the magnetic field of Jupiter induces electric currents
in this moon, which in turn create their own magnetic fields! These
fields were detected when the Galileo probe swooped closest to Europa
back in 2000:
5) M. G. Kivelson, K. K. Khurana, C. T. Russell, M. Volwerk, R.J. Walker,
and C. Zimmer, Galileo magnetometer measurements: a stronger case for a
subsurface ocean at Europa, Science, 289 (2000), 1340-1343.
At the time, Margaret Kivelson, head of the magnetometer project, said:
I think these findings tell us that there is indeed a layer of liquid
water beneath Europa's surface. I'm cautious by nature, but this new
evidence certainly makes the argument for the presence of an ocean far
more persuasive. Jupiter's magnetic field at Europa's position changes
direction every 5-1/2 hours. This changing magnetic field can drive
electrical currents in a conductor, such as an ocean. Those currents
produce a field similar to Earth's magnetic field, but with its magnetic
north pole - the location toward which a compass on Europa would point -
near Europa's equator and constantly moving. In fact, it is actually
reversing direction entirely every 5-1/2 hours.
A couple weeks ago, another nice piece of evidence was announced:
6) Paul Schenk, Isamu Matsuyama and Francis Nimmo, True polar wander
on Europa from global-scale small-circle depressions, Nature 453 (2008),
368-371.
Paul Schenk, Scars from Europa's polar wandering betray ocean beneath,
http://www.lpi.usra.edu/science/schenk/europaCropCircles/
There are two arc-shaped depressions exactly opposite each other on
Europa, each hundreds of kilometers long and between .3 and 1.5 kilometers
deep. According to the above paper, these scars have just the right shape
to be caused the moon's icy shell rotating a quarter turn relative to
the interior! The authors believe this could happen most easily if
it were floating on an ocean.
If Europa has an ocean under its ice, other questions immediately arise.
How thick is the ice and how deep is the ocean? Some guess 15-30 kilometers
of ice atop 100 kilometers of liquid. What keeps it warm? Heating
produced by tidal forces may be the best bet - radioactivity from the core
contributes just about 100 billion watts, not nearly enough:
7) M. N. Ross and G. Schubert, Tidal heating in an internal ocean model
of Europa, Nature 325 (1987), 133-144.
And then for the really big question: could there be *life* on Europa?
Antarctica has an enormous lake called Lake Vostok buried under 4
kilometers of ice, and when people drilled into it they found bizarre
life forms that had never been seen before. So, especially if Europa
had been warmer once, it's conceivable that life might have formed there
and survives to this day. Of course, the surface of Europa makes
Antarctica look downright balmy: it's -160 Celsius at the equator.
And liquid water below could be mixed with sulfuric acid, or lots of
nasty salts...
Nonetheless, some dream of sending a satellite to Europa, perhaps
to impact it at high velocity and see what's inside, or perhaps to land
and melt down through the ice:
8) Leslie Mullen, Hitting Europa hard (interview of Karl Hibbits),
Astrobiology Magazine, May 1, 2006,
http://www.astrobio.net/news/article1944.html
But these dreams may not come true anytime soon. In 2005, NASA
cancelled its ambitious plans for the Jupiter Icy Moons Orbiter:
10) Wikipedia, Jupiter Icy Moons Orbiter,
http://en.wikipedia.org/wiki/Jupiter_Icy_Moons_Orbiter
The U.S. Congress, the National Academy of Sciences, and the
NASA Advisory Committee have all supported a mission to Europa,
but NASA has still not funded this project:
11) Leonard David, Europa mission: lost in NASA budget, SPACE.com,
February 7, 2006, http://www.space.com/news/060207_europa_budget.html
Though NASA just safely landed the Phoenix spacecraft on Mars, which
is wonderful, they still spend tons of money on showy, expensive
manned missions - the Buck Rogers approach to space. So, our best
hope may lie with the European Space Agency's "Jovian Europa Orbiter",
part of a project called the Jovian Minisat Explorer:
12) ESA Science and Technology, Jovian Minisat Explorer,
http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=35982
This hasn't been funded yet, and there's no telling if it ever
will. But people are already working to make sure Europa
doesn't get contaminated by bacteria from Earth:
13) National Research Council, Preventing the Forward Contamination
of Europa, The National Academies Press, Washington, DC, 2000.
Also available at http://www.nap.edu/catalog.php?record_id=9895
In fact the US and many other countries are obligated to do this,
since they signed a United Nations treaty that requires it.
The Galileo probe had not been sterilized in a way that would
kill "extremophiles" - organisms that survive extreme conditions.
So, the National Research Council recommended that NASA crash
Galileo into Jupiter when its mission was over, to avoid an
accidental collision with Europa. So, that's what they did!
After 14 years of collecting data about Jupiter and its moons,
Galileo crashed into Jupiter and burned up in its atmosphere on
September 21, 2004.
Maybe I'll talk about other moons of Jupiter next week... the
most interesting ones besides Europa are the volcanic, sulfurous
Io and the icy Ganymede, biggest of all.
But now let me turn to the Pythagorean pentagram.
The Pythagoreans - that strange Greek cult of vegetarian
mathematicians - were apparently fascinated by the pentagram.
Why? I don't think there's any textual evidence to help us answer
this question, but luckily there's another way to settle it:
unsubstantiated wild guesses!
If you take a pentagram and keep on drawing lines through points
that are already present, you can generate this picture:
14) James Dolan, Pythagorean pentagram,
http://math.ucr.edu/home/baez/pythagorean_pentagram.jpg
This is just the beginning of an infinite picture packed with pentagrams.
The sizes of these pentagrams are related by various powers of the
golden ratio:
Phi = (sqrt(5) + 1)/2 = 1.6180339...
In particular, if you run up any arm of the big pentagram
you'll see little pentagrams, alternating blue and green
in the above picture, each 1/Phi times as big as the one before.
And if you contemplate these, you can see that:
Phi = 1 + 1/Phi
I could explain how, but I prefer to leave it as a fun little puzzle.
If you get stuck, I'll give you a clue later.
This might have interested the Pythagoreans, since it quickly implies
that
Phi = 1 + 1/Phi
= 1 + 1/(1 + 1/Phi)
= 1 + 1/(1 + 1/(1 + 1/Phi))
= 1 + 1/(1 + 1/(1 + 1/(1 + 1/Phi)))
and so on. This means that the continued fraction expansion of
Phi never ends, so it must be irrational! There's some evidence
that early Greeks were interested in continued fraction expansions...
you can read about that in this marvelous speculative book:
15) David Fowler, The Mathematics Of Plato's Academy:
A New Reconstruction, 2nd edition, Clarendon Press, Oxford, 1999.
Review by Fernando Q. Gouvêa for MAA Online available at
http://www.maa.org/reviews/mpa.html
If so, we can imagine that early Greek mathematicians discovered
the irrationality of the golden ratio by contemplating the Pythagorean
pentagram.
I recently gave a talk about this and other fun aspects of the number
5 at George Washington University and Google.
I was invited to Google by my student Mike Stay - more about that some
other day, perhaps. But I'd been invited to George Washington
University by Bill Schmitt. We went to grad school together. While I
was studying quantum field theory with Irving Segal, he was studying
combinatorics with Gian-Carlo Rota. Later he taught me about Joyal's
"especes de structures", also known as "species" or "structure types".
Later still, these turned out to be deeply related to the quantum
harmonic oscillator and Feynman diagrams! For more on that, see
"week185" and "week202".
Bill has always been interested in getting Hopf algebras from structure
types. The idea is implicit in some work of Rota:
16) Saj-Nicole Joni and Gian-Carlo Rota, Coalgebras and bialgebras in
combinatorics, Studies in Applied Mathematics 61 (1979), 93-139.
Gian-Carlo Rota, Hopf algebras in combinatorics, in Gian-Carlo
Rota on Combinatorics: Introductory Papers and Commentaries, ed.
J. P. S. Kung, Birkhauser, Boston, 1995.
but my favorite explanation is here:
17) William R. Schmitt, Hopf algebras of combinatorial structures,
Canadian Journal of Mathematics 45 (1993), 412-428. Also available
at http://home.gwu.edu/~wschmitt/papers/hacs.pdf
Let me sketch the simplest result in this paper! For starters, recall
that a structure type is any sort of structure you can put on finite
sets. In other words, it's a functor
F: FinSet_0 -> Set
where FinSet_0 is the groupoid of finite sets and bijections.
The idea is that for any finite set X, F(X) is the set all of
structures of the given type that we can put on X. A good example
is F(X) = 2^X, the set of 2-colorings of X.
Starting from this, we can form a groupoid of F-structured finite sets
and structure-preserving bijections. For example, the groupoid of
2-colored finite sets and color-preserving bijections. The idea
should be obvious, but it's good to make it precise. For category
hotshots it's just the groupoid of "elements" of F, called elt(F).
But if you're not a hotshot yet, I should explain this.
An object of elt(F) is a finite set X together with an element a in
F(X). A morphism of elt(F), say
f: (X,a) -> (X',a')
is a bijection
f: X -> X'
such that
F(f)(a) = a'
In other words: f is a bijection that carries the F-structure on X
to the F-structure on X'.
Anyway: given a structure type F, we can form a vector space B_F
whose basis consists of isomorphism classes of elt(F). And in the
paper above, Bill describes various ways to make B_F into various kinds
of coalgebra or Hopf algebra.
I'll only explain the simplest one. There are lots of structure types
where you can "restrict" a structure on a big set to a structure on a
smaller set. For example, a 2-coloring of a set restricts to a
2-coloring of any subset. Let's call such a thing a "structure type
with restriction".
Technically, a structure type with restriction is a functor
F: Inj^{op} -> Set
where Inj is the category of finite sets and injections. When
we have such a thing, the inclusion
i: X -> X'
of a little set X in a bigger set X' gives a map
F(i): F(X') -> F(X)
that says how to restrict F-structures on X' to F-structures
on X.
In this situation, Bill shows that the vector space B_F becomes
a cocommutative coalgebra. In particular, it gets a comultiplication
Delta: B_F -> B_F tensor B_F
which satisfies laws just like the commutative and associative laws
for ordinary multiplication, only "backwards".
The idea is simple: we comultiply a finite set with an F-structure on it
by chopping the set in two parts in all possible ways and using our
ability to restrict the F-structure to each part. I could write down
the formula, but it's better to guess it and then check your guess in
Bill's paper! See his Proposition 3.1.
After Bill came up with this stuff, the connection between Hopf algebras
and combinatorics became a big business - largely due to Kreimer's work
on Hopf algebras and Feynman diagrams. I talked about this back in
"week122" - but here's a more recent review, with a hundred references
for further study:
1 Kurusch Ebrahimi-Fard and Dirk Kreimer, Hopf algebra approach
to Feynman diagram calculations, available as arXiv:hep-th/0510202.
This yields lots of applications of Bill's ideas to quantum physics.
I have no idea how this huge industry is related to my work with James
Dolan and Jeffrey Morton on structure types, more general "stuff
types", quantum field theory and Feynman diagrams. But, maybe you
can figure it out if you read these:
19) John Baez and Derek Wise, Quantization and Categorification.
Fall 2003 notes: http://math.ucr.edu/home/baez/qg-fall2003
Winter 2004 notes: http://math.ucr.edu/home/baez/qg-winter2004/
Spring 2004 notes: http://math.ucr.edu/home/baez/qg-spring2004/
20) Jeffrey Morton, Categorified algebra and quantum mechanics,
Theory and Applications of Categories 16 (2006), 785-854.
Available at http://www.emis.de/journals/TAC/volumes/16/29/16-29abs.html
and as arXiv:math/0601458.
While you're mulling over these ideas, it might pay to ponder
this paper Bill told me about:
21) Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species
and Hopf algebras, available at http://www.math.tamu.edu/~maguiar/a.pdf
It's 588 pages long! It's a bunch of very sophisticated combinatorics
touching on ideas dear to my heart: q-deformation, species, Fock space,
and higher categories. I can't summarize it, but here are some
immediately gripping portions:
Chapter 5, "Higher monoidal categories". Here they discuss
"n-monoidal categories", which are categories equipped with a list
of tensor products with lax interchange laws relating each tensor
product to all the later ones on the list:
(A tensor_i B) tensor_j (A' tensor_i B') ->
(A tensor_j A') tensor_i (B tensor_j B')
for i < j. These gadgets generalize the "iterated monoidal categories"
of Balteanu, Fiedorowicz, Schwaenzel, Vogt and also Forcey - I gave some
references on these back in "week209". The big difference seems to be
that the Fiederowicz gang has all the tensor products share the same
unit. That's great for what they want to do - namely, get a kind of
category whose nerve is an n-fold loop space. But, Aguiar and Mahajan
study a bunch of examples coming from combinatorics where different
products have different units! It's really these examples that are
interesting to me, though the abstract concepts are cool too.
Chapter 7, "Hopf monoids in species". Here they use "species" to
mean what I'd call "linear structure types", that is, functors
F: FinSet_0 -> Vect
where Vect is the category of vector spaces. In section 7.9 they
take Bill Schmitt's trick for getting cocommutative coalgebras from
structure types with restriction, and use it to get cococommutative
comonoids in the category of linear structure types! In Section 7.10
they take another trick to get coalgebras from structure types:
22) William R. Schmitt, Incidence Hopf algebras, Journal of Pure and
Applied Algebra 96 (1994), 299-330. Also available at
http://home.gwu.edu/~wschmitt/papers/iha.pdf
and do something similar with that.
Chapter 9, "From species to graded vector spaces: Fock functors".
This studies what happens when you turn a Hopf monoid in the
category of linear structure types into a graded Hopf algebra -
a kind of generalized Fock space.
Chapter 11, "Hopf monoids from geometry". Here they get Hopf
monoids from the A_n Coxeter complexes, using a lot of ideas related
to Jacques Tits' theory of buildings. There's a lot of q-deformation
going on here! All these ideas are close to my heart.
You can get more of a sense of what Aguiar is up to by looking at
his homepage. I'll just list a *few* of the cool papers there:
23) Marcelo Aguiar's homepage, http://www.math.tamu.edu/~maguiar/
Marcelo Aguiar, Internal categories and quantum groups, Ph.D. thesis,
Cornell University, August 1997. Available at
http://www.math.tamu.edu/~maguiar/thesis2.pdf
Marcelo Aguiar, Braids, q-binomials and quantum groups, Advances in
Applied Mathematics 20 (1998) 323-365. Also available at
http://www.math.tamu.edu/~maguiar/braids.ps.gz
Marcelo Aguiar and Swapneel Mahajan, Coxeter groups and Hopf
algebras, Fields Institute Monographs, Volume 23, AMS, Providence, RI,
2006. Also available at http://www.math.tamu.edu/~maguiar/monograph.pdf
I was going to say a bit about quaternionic analysis, but now I'm
worn out. So, I'll just say that anyone interested in generalizing
complex analysis to the quaternions must read two papers. The first
I had managed to lose for a long time... but now I've found it again:
24) Anthony Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil.
Soc. 85 (1979), 199-225. Available at
http://citeseer.ist.psu.edu/10590.html and (slightly different version)
http://theworld.com/~sweetser/quaternions/ps/Quaternionic-analysis.pdf
The second was brought to my attention by David Corfield:
25) Igor Frenkel and Matvei Libine, Quaternionic analysis,
representation theory and physics, available as arXiv:0711.2699.
Since Igor Frenkel is a bigshot, this paper may finally bring this
neglected subject some of the attention it deserves! Like Corfield,
I'll just quote the abstract, to make your mouth water:
We develop quaternionic analysis using as a guiding principle
representation theory of various real forms of the conformal
group. We first review the Cauchy-Fueter and Poisson formulas
and explain their representation theoretic meaning. The
requirement of unitarity of representations leads us to the
extensions of these formulas in Minkowski space, which can
be viewed as another real form of quaternions. Representation
theory also suggests a quaternionic version of the Cauchy formula
for the second order pole. Remarkably, the derivative appearing
in the complex case is replaced by the Maxwell equations in the
quaternionic counterpart. We also uncover the connection between
quaternionic analysis and various structures in quantum mechanics
and quantum field theory, such as the spectrum of the hydrogen atom,
polarization of vacuum, and one-loop Feynman integrals. We also
make some further conjectures. The main goal of this and our
subsequent paper is to revive quaternionic analysis and to show
profound relations between quaternionic analysis, representation
theory and four-dimensional physics.
Finally, here's a clue for the Pythagorean pentagram puzzle. To
prove that
Phi = 1 + 1/Phi,
show the length of the longest red interval here is the sum of the
lengths of the two shorter ones:
26) James Dolan and John Baez, annotated picture of Pythagorean
pentagram, http://math.ucr.edu/home/baez/golden_ratio_pentagram.jpg
For more on the golden ratio, try "week203". For more on its
relation to the dodecahedron, see "week241".
-----------------------------------------------------------------------
Quote of the Week:
There is geometry in the humming of the strings, there is music in
the spacing of the spheres. - Pythagoras
-----------------------------------------------------------------------
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 jumping-off 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 |
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