May 27, 2007

This Week's Finds in Mathematical Physics (Week 252)

John Baez

Today I want to tell you about the electromagnetic snake at the center of our Galaxy, and continue the Tale of Groupoidification.

But first, the long-range weather forecast. There's a 40% chance of rain on Neptune in 8 billion years! More precisely, that's the chance these authors give for the formation of an ocean on Neptune when its interior cools down enough:

1) Sloane J. Wiktorowica and Andrew P. Ingersoll, Liquid water oceans in ice giants, available as astro-ph/0609723.

Right now, even though Neptune is named after the Roman god of seas and has a nice blue appearance:

it's mighty dry - at least on top. Spectroscopy detects no water at all in its upper atmosphere! But that's consistent with the presence of water down below, since water vapor is a lot heavier than hydrogen and helium, which make up most of Neptune's upper atmosphere, and the pull of gravity there is mighty fierce.

In fact, scientists believe Neptune has a core of molten metal and rock surrounded by more rock, methane ice, ammonia ice, and water ice - all solid due to high pressures. The overall density of Neptune makes sense if there's a lot of water down there. But, Wiktorowica and Ingersoll argue that the planet can't have an ocean of liquid water.

Surprisingly, this is because Neptune is too hot - even though its upper atmosphere is a chilly 50 kelvin! I think their argument goes roughly like this, though I don't understand it well. If you fell down through the atmosphere of Neptune, you'd find that it gets hotter and hotter as you go down, and moister and moister too - but always too hot, given the amount of moisture, for liquid water to be more stable than water vapor.

However, they also indulge in some predictions about the far future!

First let's set the stage:

But then, about 6.9 billion years from now, the Sun will start pulsating, ejecting half of its mass in the form of solar wind! It'll become what they call a "planetary nebula". Eventually only its inner core will be left. In "week223" I quoted Bruce Balick's eloquent description:
The remnant Sun will rise as a dot of intense light, no larger than Venus, more brilliant than 100 present Suns, and an intensely hot blue-white color hotter than any welder's torch. Light from the fiendish blue "pinprick" will braise the Earth and tear apart its surface molecules and atoms. A new but very thin "atmosphere" of free electrons will form as the Earth's surface turns to dust.

This is where Wiktorovika and Ingersoll begin their story. So far, Neptune will have warmed up a lot - assuming for the sake of argument that it wasn't thrown out of the Solar System when the Milky Way hit Andromeda. But when the Sun loses mass, Neptune will either collide with Uranus, be ejected from the Solar System, or assume a stable orbit about twice its current size.

In the latter two scenarios, Neptune will chill out, at least when the remnant Sun cools down and becomes a white dwarf. When the surface temperature of Neptune reaches 30 kelvin, they estimate it has a 41.5% plus or minus 4.2% chance of forming oceans.

So, they say: "Billions of years from now, after the Sun has gone, Neptune may therefore become the only object in the Solar system with liquid water oceans".

This sounds nice - but don't buy your beachfront property there yet.

First, they don't study how the atmospheric composition of Neptune may change when the Sun gets 3000 times brighter than now! Maybe this will help Neptune form oceans. After all, light gases like molecular hydrogen and helium, which dominate Neptune's upper atmosphere now, are more likely than water vapor to be driven off into outer space when it gets hot. But, they don't even check to see if Neptune will have any atmosphere left after this era.

Second, the estimate of a "41.5% plus or minus 4.2% chance" seems strangely precise, given the uncertainties involved. The error bars should probably have big error bars themselves!

Third, it's worth admitting that the atmosphere of Neptune is a bit mysterious. For example, nobody knows why it's bright blue. It's probably because of methane - but Uranus also has methane in its atmosphere, and it's not as blue:

Also, despite being the coldest planet in our Solar System, Neptune has the fastest winds: up to 2100 kilometers per hour, almost supersonic! Nobody knows what powers these winds.

When the Voyager 2 spacecraft flew by Neptune in 1992, it saw a storm system the size of Eurasia, which was dubbed "The Great Dark Spot":

It seemed to resemble the Great Red Spot on Jupiter, which has been around for at least 177 years. But when the Hubble Space Telescope took another good look at Neptune in 1994, the Great Dark Spot was gone! Meanwhile, other storms had formed.

So, the weather on Neptune is dynamic and poorly understood. Doing forecasts for the next 8 billion years seems pretty risky... though fun.

Of course, liquid water oceans are nice if you're looking for life. And while there probably isn't life on Neptune, there could be life on similar planets in other solar systems. So far people have found 233 of these "exoplanets", most of them heavier than Jupiter but close to their suns - because such planets are the easiest to detect by how they pull on their sun.

Here's a chart showing the masses and orbital radii of known exoplanets as of 2004:

2) P. R. Lawson, S. C. Unwin, and C. A. Beichman, Precursor Science for the Terrestrial Planet Finder, JPL Pub 04-014, Oct. 2004, page 21, fig. 5. Chart at
Report at

For comparison, the letters V, E, M, J, S, U, N, stand for planets in our Solar System, not counting Mercury or the subsequently dethroned Pluto.

As you'll see, there are lots of "hot Jupiters" - planets as big as Jupiter, or even up to 200 times heavier, but closer to their sun than we are to ours. Recently a postdoc at UCLA found evidence for water in the atmosphere of one of these planets:

3) Travis Barman, Identification of absorption features in an extrasolar planet atmosphere, available as arxiv:0704.1114.

This planet is only 7 million kilometers away from its yellow-white Sun, much closer than Mercury is to ours. Its year is only 3.5 of our days! It's bigger in size than Jupiter, but only 0.7 times as heavy. Its surface temperature must be about 1000 kelvin. That's one really hot Jupiter.

People have also been finding "hot Neptunes". In fact, I read about a nice one in the newspaper while writing this! It's called Gliese 436 b. It's the size of Neptune and it's orbiting the red dwarf star Gliese 436, which is 33 light-years from Earth. This star is only 1% as bright as our Sun - but the planet is so close that its year lasts less than three of our days! So, its surface temperature is high: higher than the melting point of lead.

However, because this planet is so big, the pressure down below can still make water into a solid. In fact, its density suggests that it's mainly made of ice!

I think this is the paper that triggered the newspaper reports:

4) M. Gillon et al, Detection of transits of the nearby hot Neptune GJ 436 b, available as arxiv:0705.2219.

It seems hot Neptunes like this could have started as hot Jupiters and then lost a lot of their atmosphere:

5) I. Baraffe, G. Chabrier, T. S. Barman, F. Selsis, F. Allard, and P. H. Hauschildt, Hot-Jupiters and hot-Neptunes, a common origin?, available at astro-ph/0505054.

There could also be cold Neptunes, perhaps with liquid water oceans. We haven't seen these yet, but they'd be hard to see. So, while Wiktorowica and Ingersoll's paper doesn't convince me about the future of our Neptune, it raises some interesting possibilities.

Next: the snake at the center of our galaxy!

Gregory Benford is best known for his science fiction, which spans the galaxy, but he's also an astrophysicist at U. C. Irvine. Recently he spent a week at my school, U. C. Riverside, which has one of the world's best SF libraries: the Eaton Collection. Since my wife is involved in the SF program at the comparative literature department here, he had dinner at our place one night. The conversation drifted all over the place, with a heavy focus on fruit flies that have been bred to live twice as long as usual. But when I asked him about his research, he said he'd written some papers about enormous glowing filaments near the center of the Milky Way.

I hadn't known about these! He said the biggest one could be a million years old, perhaps formed by some energetic event, maybe a star falling into the central black hole. Here's an expository paper he wrote about it:

6) Gregory Benford, The electromagnetic snake at the galactic center,

I'll just quote a little:

Five years ago radio astronomy revealed the oddest and longest filament yet discovered at our galactic center: a uniquely kinked structure about 150 light years long and two to three light years wide - the Snake. Its large kinks are its brightest parts. There is energetic activity at one end and a supernova bubble at the other, which the Snake appears to penetrate unharmed.

How does nature form stable, long-lived magnetic structures which display considerable polarization (about 60% at 10.55 GHz in the Snake)? In 1988 I had modeled others of the dozens of filaments seen uniquely at the galactic center in terms of an electrodynamic view, in which currents set up coherent magnetic pinches. Such self-organizing filaments can exist in laboratory plasmas for long times; the galactic ones could be at least a million years old, as estimated by the time that shear forces would disrupt them.

The electrodynamic view uses pinch forces of currents to form filaments, driven by the E = v × B of conducting molecular clouds moving across a strong milliGauss ambient, ordered field. A return current must then flow at larger radii, making a closed loop which has a springy flexibility, able to withstand the turbulent velocity fields known near the galactic center. The picture then anticipates that aberrant molecular clouds, moving contrary to the general galactic rotation, should accompany each filament. This prediction has held up as more filaments were found.

I'd like to learn more about these!

Back in "week248", I mentioned some of the complex things that electromagnetic fields and plasma do in the Sun. The center of the Galaxy is another place where electromagnetohydrodynamics runs rampant. There's a black hole there, of course, but also these filaments, and a fairly strong magnetic field that contains about 4 × 1047 joules of energy within about 300 light years of the galactic center. By comparison, a supernova emits a mere 1044 joules. So, there's a lot of energy around....

The big picture here, created by Farhad Yusef-Zadeh and collaborators, shows the galactic center quite nicely, as viewed in radio frequencies:

7) National Radio Astronomy Observatory, Origin of enigmatic galactic-center filaments revealed,

You can see some supernova remnants (SNRs), the region Sgr A which contains the black hole at the galactic center, various nonthermal radio filaments (NRFs), and the Snake. Some of these filaments come from regions where stars are forming... that could be important.

But, this article discusses another piece of the puzzle - the possible role of turbulence in winding up the galactic magnetic field:

8) Stanislav Boldyrev and Farhad Yusef-Zadeh, Turbulent origin of the galactic-center magnetic field: nonthermal radio filaments, available as astro-ph/0512373.

It's a complicated stew. I don't hope to understand it, just admire it.

And finally: the Tale of Groupoidification! In "week250" we reached the point of seeing how spans of groupoids over a fixed group G subsume the theory of G-sets and invariant relations between these - which are traditionally studied using "double cosets".

There is a lot more we could say about this. But, our most urgent goal is to see how spans of groupoids act like twice categorified matrices - matrices whose entries are not just numbers, and not just sets, but groupoids! This will expose the secret combinatorial underpinnings of a lot of fancy linear algebra. Once we've categorified linear algebra this way, we'll be in a great position to tackle fashionable topics like categorified quantum groups, invariants of higher-dimensional knots, and the like.

But, we should restrain ourselves from charging ahead too fast! Everything will hang together better if we lay the groundwork properly. For this, it pays to re-examine the history of mathematics a bit. If we're trying to understand linear algebra using groupoids, it pays to ask: how did people connect linear algebra and group theory in the first place?

This book is very helpful:

9) Charles W. Curtis, Pioneers of Representation Theory: Frobenius, Burnside, Schur and Brauer, History of Mathematics vol. 15, AMS, Providence, Rhode Island, 1999.

Back in 1897, a mathematician named William Burnside wrote the first book in English on finite groups. It was called Theory of Groups of Finite Order.

In the preface, Burnside explained why he studied finite groups by letting them act as permutations of sets, but not as linear transformations of vector spaces:

Cayley's dictum that "a group is defined by means of the laws of combination of its symbols" would imply that, in dealing with the theory of groups, no more concrete mode of representation should be used than is absolutely necessary. It may then be asked why, in a book that professes to leave all applications to one side, a considerable space is devoted to substitution groups [permutation groups], but other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could most directly be obtained by the consideration of groups of linear transformations.

In short, he didn't see the point of representing groups on vector spaces - at least as a tool in the "pure" theory of finite groups, as opposed to their applications.

However, within months after this book was published, he discovered the work of Georg Frobenius, who used linear algebra very effectively to study groups!

So, Burnside started using linear algebra in his own work on finite groups, and by the time he wrote the second edition of his book in 1911, he'd changed his tune completely:

Very considerable advances in the theory of groups of finite order have been made since the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. In fact it is now more true to say that for further advances in the abstract theory one must look largely to the representation of a group as a group of linear transformations.
It's interesting to see exactly how representing finite groups on vector spaces lets us understand them better. By now almost everyone agrees this is true. But how much of the detailed machinery of linear algebra is really needed? How much we could do purely combinatorially, using just spans of groupoids?

I don't really know the full answer to this question. But, it quickly leads us into the fascinating theory of "Hecke operators", which will play a big role in the Tale of Groupoidification. So, let's pursue it a bit.

Suppose we have two guys, William and Georg, who are studying a finite group G.

William says, "I'm studying how G acts on sets."

Georg replies, "I'm studying how it acts on complex vector spaces, as linear transformations. Mere sets are feeble entities. I can do anything you can do - but I have the tools of linear algebra to help me!"

William says, "But, you're paying a moral price. You're getting the complex numbers - a complicated infinite thing - involved in what should be a completely finitary and combinatorial subject: the study of a finite group. Is this really necessary?"

Georg replies, "I don't know, but it's sure nice. For example, suppose I have G acting on vector space V. Then I can always break down V into a direct sum of subspaces preserved by G, which can't themselves be broken down any further. In technical terms: every representation of G is a direct sum of irreducible representations. And, this decomposition is unique! It's very nice: it's a lot like how every natural integer has a unique prime factorization."

William says, "Yes, it's always fun when we can break things down into 'atoms' that can't be further decomposed. It's very satisfying to our reductionist instincts. But, I can do even better than you!"

Georg raises an eyebrow. "Oh?"

"Yeah," William says. "Suppose I have our group G acting on a set S. Then I can always break down S into a disjoint union of subsets preserved by G, which can't themselves be broken down any further. In technical terms: every action of G is a disjoint union of transitive actions. And, this decomposition is unique!"

Embarrassed, Georg replies, "Oh, right - we heard that back in "week249". I wasn't paying enough attention. But how is what you're doing better than what I'm doing? It sounds just the same."

William hesitates. "Well, first of all, a rather minor point, which I can't resist mentioning ... when you said your decomposition of representations into irreducibles was unique, you were exaggerating a little. It's just unique up to isomorphism, and not a canonical isomorphism either.

For example, if you have an irreducible representation of G on V, there are lots of different ways to slice the direct sum V + V into two copies of the representation V. It's a sort of floppy business. On the other hand, when I have a transitive action of G on S, there's exactly one way to chop the disjoint union S + S into two copies of the G-set S. I just look at the two orbits."

Georg says, "Hmm. This is actually a rather delicate point. There's not really a canonical isomorphism in your case either, since S may be isomorphic to itself in lots of ways, even as a G-set. There's something in what you say, but it's a bit hard to make precise, and it's certainly not enough to worry me."

"Okay, but here's my more serious point. Given that I can break apart any set on which G acts into 'atoms', my job is to classify those atoms: the transitive G-sets. And there's a very nice classification! Any subgroup H of G gives a transitive G-set, namely G/H, and all transitive G-sets look like this. More precisely: isomorphism classes of transitive G-sets correspond to conjugacy classes of subgroups of G.

Even better, this has a nice meaning in terms of Klein geometry. Any type of figure in Klein geometry gives a transitive G-set, with H being the stabilizer of a chosen figure of that type.

You, on the other hand, need to classify irreducible representations of G. And this is not so conceptual. What do these irreducible representations mean in terms of the group G?"

Georg replies, "Well, there's one irreducible representation for each conjugacy class in G..."

At this, William pounces. "You mean the number of isomorphism classes of irreducible representations of G equals the number of conjugacy classes in G! But as you know full well, there's no god-given correspondence. You can't just take a conjugacy class in G and cook up an irreducible representation, or irrep. So, you've just made my point. You've shown how mysterious these irreps actually are!"

Georg replies, "Actually in some cases there is a nice way to read off irreps from conjugacy classes. For example, you can do it for the symmetric groups Sn. But, I admit you can't in general... or at least, I don't know how."

William laughs, "So, I win!"

"Not at all!" replies Georg. "First, there are lots of theorems about groups that I can prove using representations, which you can't prove using actions on sets. For example, nobody knows how to prove that every group with an odd number of elements is solvable without using the tools of linear algebra."

William nods. "I admit that linear algebra is very practical. But just give us time! I proved back in 1904 that every group of size paqb is solvable if p and q are prime. To do it, I broke down and used linear algebra. But then, in 1972, Helmut Bender found a proof that avoids linear algebra."

Georg said, "Okay, struggle on then. So far, without using linear algebra, nobody can even prove my famous theorem on 'Frobenius groups'. The details don't matter here: the point is, this is a result on group actions, which seems to need linear algebra for its proof.

But if practicality won't sway you, maybe this conceptual argument will. My atoms are more fine-grained than yours!"

"What do you mean?" asks William.

"You can decompose any action of G into 'atoms', namely transitive G-sets. Similarly, I can decompose any representation of G into one of my 'atoms', namely irreps. But, there's an obvious way to turn G-sets into representations of G, and if we do this to one of your atoms, we don't get one of my atoms! We can usually break it down further! So, my atoms are smaller than yours."

"How does this work, exactly?"

"It's easy," says Georg, getting a bit cocky. "Say you have a group G acting on a set S. Then I can form the vector space C[S] whose elements are formal linear combinations of elements of S. In other words, it's the vector space having S as basis. If we're feeling sloppy we can think of guys in C[S] as functions on S that vanish except at finitely many points. It's better to think of them as measures on S. But anyway: since G acts on S, it acts linearly on C[S]!

So, any G-set gives a representation of G. But, even when G acts transitively on S, its representation on C[S] is hardly ever irreducible."

William glowers. "Oh yeah?"

"Yeah. Suppose for example that S is finite. Then the constant functions on S form a 1-dimensional subspace of C[S] that's invariant under the action of G. So, at the very least, we can break C[S] into two pieces."

"Well," replies William defensively, "That's pretty obvious. But it's also not such a big deal. So you can break up any my transitive G-sets into two of your irreps, one being the 'trivial' irrep. So what???"

"It wouldn't be a big deal if that's all that ever happened," says Georg. "In fact, we can break C[S] into precisely two irreps whenever the action of G on S is 'doubly transitive' - meaning we can send any pair of distinct elements of S to any other using some element of G. But, there lots of transitive actions aren't doubly transitive! And usually, one your atoms breaks down into a bunch of my atoms. In fact I'd like to show you how this works, in detail, for the symmetric groups."

"Maybe next week," says William. "But, I see your point. Your atoms are more atomic than my atoms."

Georg seems to have won the argument. But, William wouldn't have conceded the point quite so fast, if he'd thought about invariant relations!

The point is this. Suppose we have two G-sets, say X and Y. Any G-set map from X to Y gives an intertwining operator from C[X] to C[Y]. But, even after taking linear combinations, there are typically plenty of intertwining operators that don't arise this way. It's these extra intertwining operators that let us chop representations into smaller atoms.

But where do these extra intertwining operators come from? They come from invariant relations between X and Y!

And, what are these extra intertwining operators called? In some famous special cases, like in study of modular forms, they're called "Hecke operators". In some other famous special cases, like in the study of symmetric groups, they form algebras called "Hecke algebras".

A lot of people don't even know that Hecke operators and Hecke algebras are two faces of the same idea: getting intertwining operators from invariant relations. But, we'll see this is true, once we look at some examples.

I think I'll save those for future episodes. But if you've followed the Tale so far, you can probably stand a few extra hints of where we're going. Recall from "week250" that invariant relations between G-sets are just spans of groupoids equipped with some extra stuff. So, invariant relations between G-sets are just a warmup for the more general, and simpler, theory of spans of groupoids. I said back in "week248" that spans of groupoids give linear operators. What I'm trying to say now is that these linear operators are a massive generalization - but also a simplification - of what people call "Hecke operators".

Finally, for students trying to learn a little basic category theory, I'd like to cast the argument between William and Georg into the language of categories, just to help you practice your vocabulary.

A G-set is the same as a functor

A: G → Set

where we think of G as a 1-object category. There's a category of G-sets, namely


This has functors A: G → Set as objects, and natural transformations between these as morphisms. Usually the objects are called "G-sets", and the morphisms are called "maps of G-sets".

We can also play this whole game with the category of vector spaces replacing the category of sets. A representation of G is the same as a functor

A: G → Vect

As before, there's a category of such things, namely


This has functors A: G → Vect as objects, and natural transformations between these as morphisms. Now the objects are called "representations of G" and the morphisms are called "intertwining operators".

We could let any groupoid take the place of the group G. We could also let any other category take the place of Set or Vect.

Part of what William and Georg were debating was: how similar are hom(G,Set) and hom(G,Vect)? How are they related?

First of all, there's a functor

F: Set → Vect

sending each set S to the vector space C[S] with that set as basis. So, given an action of G on a set:

A: G → Set

we can compose it with F and get a representation of G:

FA: G → Vect

This kind of representation is called a "permutation representation". And, this trick gives a functor from G-sets to representations of G:

hom(G,Set) → hom(G,Vect) 
        A |-> FA
If this functor were an equivalence of categories, it would have to be essentially surjective, full and faithful. But, not every representation of G is isomorphic to a permutation representation! In other words, the functor

hom(G,Set) → hom(G,Vect)

is not "essentially surjective".

Moreover, not every intertwining operator between permutation representations comes from a map between their underlying G-sets! In other words, the functor

hom(G,Set) → hom(G,Vect)

is not "full".

But, given two different maps from one G-set to another, they give different intertwining operators. So, at least our functor is "faithful".

Maps of G-sets are a special case of invariant relations. So, to get a category that more closely resembles hom(G,Vect), while remaining purely combinatorial, we can replace hom(G,Set) by the category with G-sets as objects and invariant binary relations as morphisms. This is the basic idea of "Hecke operators".

Or, even better, we can try a weak 2-category, with

This is where groupoidification comes into its own.

Addendum: For more discussion, go to the n-Category Café.

The present treatise is intended to introduce to the reader the main outlines of the theory of groups of finite order apart from any applications. The subject is one which has hitherto attracted but little attention in this country; it will afford me much satisfaction if, by means of this book, I shall arouse interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied - William Burnside

© 2007 John Baez