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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 [Broken]

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:

18) 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 [Broken]

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/ [Broken]

Marcelo Aguiar, Internal categories and quantum groups, Ph.D. thesis,

Cornell University, August 1997. Available at

http://www.math.tamu.edu/~maguiar/thesis2.pdf [Broken]

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 [Broken]

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 [Broken]

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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