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March 28, 2007

This Week's Finds in Mathematical Physics (Week 248)

John Baez

This week I'll continue the Tale of Groupoidification, but first -

relativity on the world-wide web, and some new views of the Sun!

Chris Hillman has always been one of the most erudite and enigmatic

explainers of mathematical physics on the internet, from the early

days of sci.physics, to sci.physics.research, to the rise of Wikipedia.

I know him fairly well, but I've never actually met him. Feared

by crackpots worldwide, some claim he is a "software agent" - an

artificial intelligence run amuck. He has never denied this; in

fact, I'm beginning to believe it's true.

Anyway, he has just updated his wonderful guide to relativity:

1) Chris Hillman, Relativity on the World-Wide Web,

http://math.ucr.edu/home/baez/RelWWW/

Regardless of where you stand on the road to knowledge - whether

you just want to see cool animations of black holes, or need software

for doing tensor calculations, or want to learn more about advanced

astrophysics - this has something for you!

Speaking of astrophysics - here's a cool movie of the Moon passing

in front of the Sun, as viewed from the "STEREO B" spacecraft:

2) Astronomy Picture of the Day, March 3 2007, Lunar transit from STEREO,

http://antwrp.gsfc.nasa.gov/apod/ap070303.html

As the name hints, there's a pair of STEREO satellites in orbit around

the Sun. One is leading the Earth a little, the other lagging behind a

bit, to provide a stereoscopic view of coronal mass ejections.

What's a "coronal mass ejection"? It's an event where the Sun shoots

off a blob of ionized gas - billions of tons of it - at speeds around

1000 kilometers per second!

That sounds cataclysmic... but it happens between once a day and

5-6 times a day, depending on where we are in the 11-year solar

cycle, also known as the "sunspot cycle". Right now we're near

the minimum of this cycle. Near the maximum, coronal mass ejections

can really screw up communication systems here on Earth. For

example, in 1998 a big one seems to have knocked out a communication

satellite called Galaxy 4, causing 45 million people in the US to

lose their telephone pager service:

3) Gordon Holman and Sarah Benedict, Solar Flare Theory:

Coronal mass ejections, solar flares, and the Earth-Sun connection,

http://www.agu.org/sci_soc/articles/eisbaker.html

So, it's not only fun but also practical to understand coronal

mass ejections. Here's a movie of one taken by the Solar and

Heliospheric observatory (SOHO):

4) NASA, Cannibal coronal mass ejections,

http://science.nasa.gov/headlines/y2001/ast27mar_1.htm [Broken]

As I mentioned in "week150", SOHO is a satellite orbiting the

Sun right in front of the Earth, at an unstable equilibrium -

a "Lagrange point" - called L1. SOHO is bristling with detectors

and telescopes of all sorts, and this movie was taken by a coronagraph,

which is a telescope specially designed to block out the Sun's

disk and see the fainter corona.

If a coronal mass ejection hits the Earth, it does something like this:

5) NASA, What is a CME?,

http://www.nasa.gov/mpg/111836main_what_is_a_cme_NASA WebV_1.mpg

In this artist's depiction you can see the plasma shoot off from the

Sun, hit the Earth's magnetic field - this actually takes one to five

days - and squash it, pushing field lines around to the back side of

the Earth. When the magnetic field lines reconnect in back, trillions

of watts of power come cascading down through the upper atmosphere,

producing auroras. Here's a nice movie of what *those* can look like:

6) YouTube, Aurora (Northern Lights),

I wish I understood this magnetic field line trickery better!

Magnetohydrodynamics - the interactions between electromagnetic fields

and plasma - is a branch of physics that always gave me the shivers.

The Navier-Stokes equations describing fluid flow are bad enough - if

you can prove they have solutions, you'll win $1,000,000 from the Clay

Mathematics Institute. Throw in Maxwell's equations and you get a

real witches' brew of strange phenomena.

In fact, this subject is puzzling even to experts. For example,

why is the Sun's upper atmosphere - the corona - so hot? Here's

a picture of the Sun in X-rays taken by another satellite:

7) Transition Region and Coronal Explorer (TRACE), Images of the sun,

http://trace.lmsal.com/POD/TRACEpodarchive26.html

This lets you see plasma in the corona with temperatures between

1 million kelvin (shown as blue) and 2 million kelvin (red). By

comparison, the visible surface of the Sun is a mere 5800 kelvin!

Where does the energy come from to heat the corona? There are lots

of competing theories. I bet interactions between the Sun's magnetic

fields and plasma are the answer - but they're incredibly complicated.

A new satellite called Hinode recently got a good look at what's going

on, and it seems the magnetic field on the Sun's surface is much more

dynamic than before thought:

8) NASA, Hinode: investigating the Sun's magnetic field,

http://www.nasa.gov/mission_pages/solar-b/

In fact, weather on the Sun may be more complex than on the Earth.

There's "rain" when plasma from the corona cools and falls back down

to the Sun's surface... and sometimes there are even tornados! You

think tornados on Earth are scary? Check out this movie made during

an 8-hour period in August 2000, near the height of the solar cycle:

9) TRACE, Tornados and fountains in a filament on 2 Aug. 2000,

movie 13, http://trace.lmsal.com/POD/

Besides the tornados, near the end you can see glowing filaments of plasma

following magnetic field lines!

Now for something simpler: the Tale of Groupoidification.

I don't want this to be accessible only to experts, since a bunch of

it is so wonderfully elementary. So, I'm going to proceed rather slowly.

This may make the experts impatient, so near the end I'll zip ahead and

sketch out a bit of the big picture.

Last time I introduced spans of sets. A span of sets is just a set S

equipped with functions to X and Y:

S

/ \

/ \

F/ \G

/ \

v v

X Y

Simple! But the important thing is to understand this thing as a

"witnessed relation".

Have you heard how computer scientists use the term "witness"? They

say the number 17 is a "witness" to the fact that the number 221 isn't

prime, since 17 evenly divides 221.

That's the idea here. Given a span S as above, we can say an element

x of X and an element y of Y are "related" if there's an element s of S

with

F(s) = x and G(s) = y.

The element s is a "witness" to the relation.

Last week, I gave an example where a Frenchman x and an Englishwoman y

were related if they were both the favorites of some Russian s.

Note: there's more information in the span than the relation it

determines. The relation either holds or fails to hold. The span

does more: it provides a set of "witnesses". The relation holds

if this set of witnesses is nonempty, and fails to hold if it's empty.

At least, that's how mathematicians think. When I got married last

month, we discovered the state of California demands TWO witnesses

attend the ceremony and sign the application for a marriage license.

Here the relation is "being married", and the witnesses attest to that

relation - but for the state, one witness is not enough to prove that

the relation holds! They're using a more cautious form of logic.

To get the really interesting math to show up, we need to look at

other examples of "witnessed relations" - not involving Russians

or marriages, but geometry and symmetry.

For example, suppose we're doing 3-dimensional geometry. There's a

relation "the point x and the line y lie on a plane", but it's pretty

dull, since it's always true. More interesting is the witnessed

relation "the point x and the line y lie on the plane z". The reason

is that sometimes there will be just one plane containing a point and

a line, but when the point lies on the line, there will be lots.

To think of this "witnessed relation" as a span

S

/ \

/ \

F/ \G

/ \

v v

X Y

we can take X to be the set of points and Y to be the set of lines.

Can we take S to be the set of planes? No! Then there would be no way

to define the functions f and g, because the same plane contains lots of

different points and lines. So, we should take S to be the set of

triples (x,y,z) where x is a point, y is a line, and z is a plane

containing x and y. Then we can take

F(x,y,z) = x

and

G(x,y,z) = y.

A "witness" to the fact that x and y lie on a plane is not just a

plane containing them, but the entire triple.

(If you're really paying attention, you'll have noticed that we need to

play the same trick in the example of witnesses to a marriage.)

Spans like this play a big role in "incidence geometry". There are

lots of flavors of incidence geometry, with "projective geometry"

being the most famous. But, a common feature is that we always have

various kinds of "figures" - like points, lines, planes, and so on.

And, we have various kinds of "incidence relations" involving these

figures. But to really understand incidence geometry, we need to

go beyond relations and use spans of sets.

Actually, we need to go beyond spans of sets and use spans of groupoids!

The reason is that incidence geometries usually have interesting symmetries,

and a groupoid is like a "set with symmetries". For example, consider

lines in 3-dimensional space. These form a set, but there are also

symmetries of 3-dimensional space mapping one line to another. To

take these into account we need a richer structure: a groupoid!

Here's the formal definition: a groupoid consists of a set of "objects",

and for any objects x and y, a set of "morphisms"

f: x -> y

which we think of as symmetries taking x to y. We can compose a morphism

f: x -> y and a morphism g: y -> z to get a morphism fg: x -> z. We

think of fg as the result of doing first f and then g. So, we demand the

associative law

(fg)h = f(gh)

whenever either side is well-defined. We also demand that every object

x has an identity morphism

1_x: x -> x

We think of this as the symmetry that doesn't do anything to x.

So, given any morphism f: x -> y, we demand that

f 1_y = f = 1_x f.

So far this is the definition of a "category". What makes it a

"groupoid" is that every morphism f: x -> y has an "inverse"

f^{-1}: y -> x

with the property that

f f^{-1} = 1_x

and

f^{-1} f = 1_y.

In other words, we can "undo" any symmetry.

So, in our spans from incidence geometry:

S

/ \

/ \

F/ \G

/ \

v v

X Y

X, Y and S will be groupoids, while F and G will be maps between

groupoids: that is, "functors"!

What's a functor? Given groupoids A and B, clearly a functor

F: A -> B

should send any object x in A to an object F(x) in B. But also, it

sends any morphism in A:

f: x -> y

to a morphism in B:

F(f): F(x) -> F(y).

And, it should preserve preserves all the structure that a groupoid has,

namely composition:

F(fg) = F(f) F(g)

and identities:

F(1_x) = 1_{F(x)}.

It then automatically preserves inverses too:

F(f^{-1}) = F(f)^{-1}

Given this, what's the meaning of a span of groupoids? You could say

it's a "invariant" witnessed relation - that is, a relation with

witnesses that's *preserved* by the symmetries at hand. These are the

very essence of incidence geometry. For example, if we have a point and

a line lying on a plane, we can rotate the whole picture and get a new

point and a new line containing a new plane. Indeed, a "symmetry" in

incidence geometry is precisely something that preserves all such

"incidence relations".

For those of you not comfy with groupoids, let's see how this actually

works. Suppose we have a span of groupoids:

S

/ \

/ \

F/ \G

/ \

v v

X Y

and the object s is a witness to the fact that x and y are related:

F(s) = x and G(s) = y.

Also suppose we have a symmetry sending s to some other object of S:

f: s -> s'

This gives morphisms

F(f): F(s) -> F(s')

in X and

G(f): G(s) -> G(s')

in Y. And if we define

F(s') = x' and G(s') = y',

we see that s' is a witness to the fact that x' and y' are related.

Let me summarize the Tale so far:

Spans of groupoids describe "invariant witnessed relations".

"Invariant witnesses relations" are the essence of incidence geometry.

There's a way to turn spans of groupoids into matrices of numbers,

so that multiplying matrices corresponds to some nice way of

"composing" spans of groupoids (which I haven't really explained yet).

>From all this, you should begin to vaguely see that starting from any sort

of incidence geometry, we should be able to get a bunch of matrices.

Facts about incidence geometry will give facts about linear algebra!

"Groupoidification" is an attempt to reverse-engineer this process.

We will discover that lots of famous facts about linear algebra are

secretly facts about incidence geometry!

To prepare for what's to come, the maniacally diligent reader might

like to review "week178", "week180", "week181", "week186" and "week187",

where I explained how any Dynkin diagram gives rise to a flavor

of incidence geometry. For example, the simplest-looking Dynkin

diagrams, the A_n series, like this for n = 3:

o------o------o

points lines planes

give rise to n-dimensional projective geometry. I may have to review

this stuff, but first I'll probably say a bit about the theory of

group representations and Hecke algebras.

(There will also be other ways to get spans of groupoids, that don't

quite fit into what's customarily called "incidence geometry", but

still fit very nicely into our Tale. For example, Dynkin diagrams

become "quivers" when we give each edge a direction, and the "groupoid

of representations of a quiver" gives rise to linear-algebraic

structures related to a quantum group. In fact, I already mentioned

this in item E of "week230". Eventually this will let us groupoidify

the whole theory of quantum groups! But, I don't want to rush into that,

since it makes more sense when put in the right context.)

By the way, some of you have already pointed out how unfortunate it is

that *last* Week was devoted to E8, instead of *this* one. Sorry.

-----------------------------------------------------------------------

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If you just want the latest issue, go to

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# This Week's Finds in Mathematical Physics (Week 248)

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