Also available at http://math.ucr.edu/home/baez/week273.html
December 14, 2008
This Week's Finds in Mathematical Physics (Week 273)
John Baez
Today I'd like to talk about the history of the Earth, and
then say a bit about locally compact abelian groups. But
first, a few more words about Enceladus.
Last week we visited the geysers of Saturn's moon Enceladus.
Afterwards, George Musser pointed me to an article on this
subject by Carolyn Porco, leader of the imaging team for the
Cassini-Huygens mission - the team that's been taking the
photos I showed you. It's a great article, leading up to some
intriguing theories about what powers these geysers:
1) Carolyn Porco, Enceladus: secrets of Saturn's strangest moon,
Scientific American, November 2008, available at
http://www.sciam.com/article.cfm?id=enceladus-secrets
And it's free online! - at least for now. I've criticized the
Scientific American before here, but if they keep coming out with
articles like this, I'll change my tune. For one thing, it's
well-written:
There is obviously a tale writ on the countenance of this
little moon that tells of dramatic events in its past, but
its present, we were about to find out, is more stunning by far.
In its excursion over the outskirts of the south polar terrain,
Cassini's dust analyzer picked up tiny particles, apparently
coming from the region of the tiger stripes. Two other
instruments detected water vapor, and one of them delivered the
signature of carbon dioxide, nitrogen and methane. Cassini
had passed through a tenuous cloud.
What is more, the thermal infrared imager sensed elevated
temperatures along the fractures - possibly as high as 180
kelvins, well above the 70 kelvins that would be expected
from simple heating by sunlight. These locales pump out an
extraordinary 60 watts per square meter, many times more than
the 2.5 watts per square meter of heat arising from Yellowstone's
geothermal area. And smaller patches of surface, beyond the
resolving power of the infrared instrument, could be even hotter.
For another, it tackles a fascinating mystery. Where does all this
power come from? The geysers near the south pole of Enceladus emit
about 6 gigawatts of heat. Enceladus is too small to have that much
radioactive heating at its core - only about 0.3 gigawatts, probably.
The rest must come from tidal heating. This happens when stuff
sloshes back and forth in a changing gravitional field: friction
converts this motion to heat.
So, what causes tides on Enceladus? It may be important that Enceladus
has a 2:1 resonance with Dione: it orbits Saturn twice for each orbit
of that larger moon. This sort of resonance is known to cause tidal
heating. For example, in "week269", I showed you how Jupiter's moon
Io is locked in resonances with Europa and Ganymede. The resulting
tidal heat powers its mighty volcanos.
Unfortunately, the resonance with Dione doesn't seem powerful enough
to produce the heat we see on Enceladus. Unless something funny is
going on, there should only be 0.1 gigawatts of tidal heating - not
nearly enough! At least that's what Porco estimated in 2006:
2) Carolyn Porco et al, Cassini observes the active south pole of
Enceladus, Science 311 (2006), 1393-1401.
So, we need to dream up a more complicated story.
Here's one: there could be a kind of slow cycle where the orbit of
Enceladus gets more eccentric, tidal heating increases, ice beneath
its surface melts, more sloshing water causes more tidal heating, and
then the release of heat energy damps its eccentric orbit, until it
freezes solid and the whole cycle starts over. We could be near the
end of such a cycle right now.
Here's another: maybe Enceladus has an sea of liquid water under the
frozen surface of its south pole. With enough water sloshing around,
there could be a lot more tidal heating than you'd naively guess...
and this heating, in turn, could keep the water liquid. The fun thing
about this second scenario is that a permanent liquid ocean on
Enceladus raises the possibility of life!
Nobody knows for sure what's going on - but Carolyn Porco examines the
options in a clear and engaging way. If you like celestial mechanics,
also try this paper:
3) Jennifer Meyer, Jack Wisdom, Tidal heating in Enceladus, Icarus 188
(2007), 535-539. Also available at
http://groups.csail.mit.edu/mac/users/wisdom/meyerwisdom1.pdf
I wrote about Jack Wisdom's work back in "week107" - it's fascinating
stuff. He knows a lot about resonances. For related work on the
Jupiter-Saturn resonance, the Neptune-Pluto resonance, and the math of
continued fractions, also try the addenda to "week222".
Next I'd like to give you a quick trip through the Earth's history.
In "week196" we looked back into the deep past, all the way to the
electroweak phase transition 10 picoseconds after the Big Bang. On
the other hand, here:
4) John Baez, The end of the universe,
http://math.ucr.edu/home/baez/end.html
you can zip forwards into the deep future - for example, 10^{19} years
from now, when the galaxies boil off, shooting dead stars into the the
vast night.
But now I'd like to zoom in closer to home and quickly tell the
history of the Earth, focusing on an aspect you may never have thought
about. You see, Kevin Kelly recently pointed me to this fascinating
paper on "mineral evolution":
5) Robert M. Hazen, Dominic Papineau, Wouter Bleeker, Robert T. Downs,
John M. Ferry, Timothy J. McCoy, Dmitri A. Sverjensky and Henxiong Yang,
Mineral evolution, American Mineralogist 91 (2008), 1693-1720.
Ever since the Earth was formed, the number of different minerals
has kept increasing - and ever since *life* ran wild, it's soared!
Some examples are obvious: seashells become limestone, which gets
squashed into marble. But others are less so.
Here's a timeline loosely taken from this paper:
THE ERA OF PLANETARY FORMATION
1. Primary chondrite minerals (over 4.56 billion years ago):
60 species of mineral.
Chondrites are stony meteorites that formed early in the history of
the solar system. They're made of chondrules - millimeter-sized
spheres of olivine, pyroxene and other minerals - together with
nuggets called "CAIs" (calcium-aluminum rich inclusions) and
other stuff. These chondrules began life as molten droplets back
when the Sun was a T Tauri star, heated only by gravitational collapse.
2. Aqueous alteration, thermal alteration, and shocks form
achondrites and iron-nickel meteorites (4.56 to 4.55 billion
years ago): 250 species of mineral.
This is the era when the disk of dust circling the early Sun started
forming lumps. As these lumps collided, they got bigger and bigger,
eventually forming the asteroids and planets we see today. Some of
these proto-planets melted, letting heavier metals sink to their core
while lighter material stayed on top. But then some crashed into each
other, shattering and forming new kinds of meteorites: iron-nickel
meteorites, and stony meteorites called "achondrites". Thanks to
radioactive dating of these, scientists claim a shockingly precise
knowledge of when all this happened: sometime between 4.56 and 4.55
billion years ago.
THE ERA OF CRUST AND MANTLE REWORKING
3. Igneous rock evolution (4.55 to 4 billion years ago):
350 species of mineral.
The Earth's history is divided into four eons: Hadean, Archean,
Proterozoic and Phanerozoic. Back when I was a kid, the Cambrian
era seemed really old - but that's just the start of the current
eon, when multicellular life emerged: the Phanerozoic. We're digging
much deeper now: the Phanerozoic will be *end* of today's story.
The Hadean eon began with a bang: the event that formed the Moon
around 4.55 billion years ago! What made the moon? The current
most popular explanation is the "giant impact theory" - sometimes
called the Big Splat Theory. Dana Mackenzie spends a lot of time
writing about math, but he's also written a book about this:
6) Dana Mackenzie, The Big Splat, or How Our Moon Came To Be,
Wiley, New York, 2003.
The idea is that another planet formed in one of the "Lagrange
points" of Earth's orbit - a stable spot 60 degrees ahead or
behind the Earth:
7) John Baez, Lagrange points,
http://math.ucr.edu/home/baez/lagrange.html
But when this planet reached about the mass of Mars, it would no
longer be stable in this location. So, it gradually drifted toward
Earth, and eventually smacked right into us! The impact was incredibly
energetic, melting the Earth's entire crust and outer mantle. The iron
core of this other planet sank into Earth's core, while about 2% of the
outer part formed an orbiting ring of debris. Within a century, about
half of this ring formed the Moon we know and love.
It's an amazing story, but most of the evidence seems to support it.
The early Moon is known to have been much closer to Earth than it
is now - it's been receding ever since. For this and many other
reasons, the giant impact theory is sufficiently solid that the
hypothetical doomed planet that hit Earth has a name: Theia! You
can even watch a simulation of it hitting Earth, produced by Robin
Canup:
8) Dana Mackenzie, The Big Splat (animation),
http://www.danamackenzie.com/big_splat_animation.htm
Let me quote Mackenzie on this:
The simulation shows the first twenty-four hours after the giant
impact. It begins with Theia about to strike the Earth. After
the impact, one hemisphere of the Earth is sheared off and flung
into space. The remaining part of Earth is very lopsided, and sets
up a "gravitational torque" on the debris. This boosts some of
the debris into orbit. (Without such a boost, it would all simply
fall back down again.)
Within a few hours, the debris has formed an "arm" that smashes
spectacularly back into the Earth. This crash is nearly as
explosive as the original impact! (The second explosion can be
seen much more vividly in the video than in the still frames
published in my book.) Notice how the temperature of the Earth
has risen, from the blues and greens of the early frames to yellows
and reds, indicating more than 2000 degrees Kelvin. Earth has
literally become a blast furnace.
As the fateful day continues, the debris gets more uniformly
distributed in a disk around the Earth. Notice, though, that
this disk is not stable like the rings of Saturn. It develops
shock waves that whirl around the Earth, collecting material
into spiral arms. According to Alastair Cameron, another modeler
the giant impact, these spiral arms also play an important role
in the development of the Moon, by "siphoning" debris up from
lower orbits into higher ones. Scientists have estimated that a
mass at least twice the present mass of the Moon had to be lifted
beyond the "Roche limit," roughly twelve thousand miles or three
Earth radii above the surface. Any debris that does not make it
past the Roche limit will be torn apart by tidal forces, and cannot
form a permanent moon.
This simulation stops after 24 hours, a long time before the disk
of debris condenses into our Moon. The Moon was not formed in a
day! However, it did form much more rapidly than you might expect;
current estimates range from 1 to 100 years. This is astounding,
compared to ordinary geological time scales. An entire new planet
was born within the life span of a single human.
No rocks on Earth are known to survive from before 4.03 billion years
ago, so the details of this time period are hotly debated. However,
many igneous rocks, especially basalt, must have been formed at this
time.
Even after the surface cooled enough to form a crust, volcanoes
continued to release steam, carbon dioxide, and ammonia. This led
to what is called the Earth's "second atmosphere". The "first
atmosphere" was mainly hydrogen and helium; the second was mainly
carbon dioxide and water vapor, with some nitrogen but almost no
oxygen. This second atmosphere had about 100 times as much gas as
today's "third atmosphere"!
As the Earth cooled, oceans formed, and much of the carbon dioxide
dissolved into the seawater and later precipitated out as carbonates.
4. Granitoid formation and the first cratons (4 to 3.2 billion years
ago): 1000 species of mineral.
Between 4 and 3.8 billion years ago there was another scary time:
the Late Heavy Bombardment. A lot of large craters on the Moon
date to this period, so probably the Earth, Venus and Mars got hit
too. Why so many impacts after a period of relative calm? One
theory is that Jupiter and Saturn locked into their current 2:1
resonance at this time, causing a big shakedown in the asteroid
belt and Kuiper belt. Wikipedia has a nice quick review of this
and other theories:
9) Wikipedia, Late heavy bombardment,
http://en.wikipedia.org/wiki/Late_heavy_bombardment
This time also marked the rise of "cratons". Cratons are a bit like
small early "plates" in the sense of plate tectonics: they're ancient
tightly-knit pieces of the earth's crust and mantle, many of which
survive today. While most cratons only finished forming 2.7 billion
years ago, nearly all started growing earlier, in the Eoarchean era.
Cratons are made largely of "granitoids". Granitoids are more
sophisticated igneous rocks than basalt. Modern granite is one
of these. Granite is made in a variety of ways, for example by
the remelting of sedimentary rock. Early granitoids were probably
simpler.
5. Emergence of plate tectonics (3.2 to 2.8 billion years ago):
1500 species of mineral.
In the Paleoarchean and Mesoarchean eras, plate tectonics as we
know it began. A key aspect of this process is the recycling of the
Earth's crust through "subduction": oceanic plates slide under
continental plates and get pushed down into the mantle. Another
feature is underwater volcanism and hydrothermal activity.
6. Anoxic biology leading up to photosynthesis (3.9 to 2.5 billion
years ago): 1500 species of mineral.
The earliest hints of life include some rocks called "banded
iron formations" that date back 3.85 billion years. The real
fun starts with the rise of photosynthesis leading up to the
Great Oxidation Event about 2.5 billion years ago - more on that
later. But organisms from the domain Archaea can do well in a
wide variety of extreme environments without oxygen, and as their
name suggests, many of these organisms are very ancient. These
organisms gave rise to an active sulfur cycle and deposits of
sulfate ores starting in the Paleoarchean era. They also made
the atmosphere increasingly rich in methane throughout the
Mesoarchean and Neoarchean. So, life was already beginning to
affect mineral evolution.
THE ERA OF BIO-MEDIATED MINERAL FORMATION
7. The Great Oxidation Event (2.5 to 1.9 billion years ago):
over 4000 species of mineral.
The Archean eon ended and the Proterozoic began with the Great
Oxidation Event 2.5 billion years ago. In this event, also known as
the Oxygen Catastrophe, photosynthesis put enough oxygen into the
atmosphere to make it lethal to most organisms of the time! Luckily
evolution found a way out of this impasee. The oyxgen-rich atmosphere
in turn led to a wide variety of new minerals.
8. The intermediate ocean (1.9 to 1 billion years ago):
over 4000 species of mineral.
In the Mesoproterozoic era, increased oxygen levels in the ocean put
an end to many anoxic life forms. For example, around 1.85 billion
years ago, banded iron formations suddenly ceased. The next gigayear
was rather static and dull - if you're mainly interested in new
minerals, that is. The term "intermediate ocean" means that during
this period, the seawater contained a lot more oxygen than before,
but still much less than today.
9. Snowball Earth and the Neoproterozoic oxygenation events
(1 to 0.54 billion years ago): over 4000 species of mineral.
The Neoproterozoic era probably saw several "Snowball Earth"
events: episodes of runaway glaciation during which most or
all the Earth was covered with ice. Since ice reflects sunlight,
making the Earth even colder, it's easy to guess how this runaway
feedback might happen. The opposite sort of feedback is happening
now, as melting ice makes the Earth darker and thus even warmer.
The interesting questions are why this instability doesn't keep
driving the Earth to extreme temperatures one way or another -
and what stopped the Snowball Earth events back then!
Here's a currently popular answer to the second question.
Ice sheets slow down the weathering of rock. Weathering of rock
is one of the main long-term processes that use up atmospheric carbon
dioxide, by converting it into various carbonate minerals. On the
other hand, even on an ice-covered Earth, volcanic activity would
keep putting carbon dioxide into the atmosphere. So, eventually
carbon dioxide would build up, and the greenhouse effect would warm
things up again. This process might be very dramatic, with perhaps
as much as 13% of the atmosphere being carbon dioxide (350 times
what we see today), and temperatures soaring to 50 Celsius!
But the details are still the subject of much controversy.
At the end of these glacial cycles, it's believed that oxygen
increased from 2% of the atmosphere to 15%. (Now it's 21%.)
This may be why multi-celled oxygen-breathing organisms date
back to this time. Others argue that the "freeze-fry" cycle
imposed tremendous evolutionary pressure on life and led to
the rise of multicellular organisms. Both these theories could
be true.
During the glacial cycles, few new minerals were formed - unless
you count ice. Afterwards, surface rocks were weathered in new
ways involving oxidation.
10. Phanerozoic biomineralization (0.54 billion years ago to now):
over 4300 species of mineral.
The Phanerozoic eon, beginning with the Cambrian 540 million years
ago, marks the rise of life as we know it. During this time, sea
life has given rise to extensive deposits of biominerals such as
calcite, aragonite, dolomite, hydroxylapatite, and opal. There has
also been increased production of clay and many different types
of soil.
This is the end of our story - but of course the story isn't over.
We're now in the Anthropocene epoch of the Cenozoic era of the
Phanerozoic eon. New things are happening. Humans are boosting
atmosphieric carbon dioxide levels. If the temperature rises one
more degree, the Earth's temperature will be the hottest it's been
in 1.35 million years, before the Ice Ages began. There's no telling
when this trend will stop. We're filling the oceans and land with
plastic and other debris. In millions of years, these may form new
species of minerals. Regardless, there will probably still be rocks -
but we'll either be gone or drastically changed.
Next: Pontryagin duality! Like last week's math topic, I needed to
learn more about this for my work on infinite-dimensional
representations of 2-groups. And like last week's math topic, it
involves a lot of analysis. But it also involves a lot of algebra
and category theory.
You may know about Fourier series, which lets you take a sufficiently
nice complex-valued function on the circle and write it like this:
f(x) = sum_k g_k exp(ikx)
Here k ranges over all integers, so what you're really doing here
is taking a function on the circle:
f: S^1 -> C
and expressing it in terms of a function on the integers:
g: Z -> C
More precisely, any L^2 function on the circle can be expressed
this way for some L^2 function on the integers - and conversely.
In fact, if we normalize things right, the Fourier series gives
a unitary isomorphism between the Hilbert spaces L^2(S^1) and L^2(Z).
You may also know about the Fourier transform, which lets you take a
sufficiently nice complex-valued function on the real line and
write it like this:
f(x) = integral g(k) exp(ink) dk
Here k also ranges over the real line, so what you're really
doing is taking a function on the line:
f: R -> C
and expressing it in terms of another function on the line:
g: R -> C
In fact, any L^2 function f: R -> C can be expressed this way
for some L^2 function g: R -> C. And if we normalize things
right, the Fourier transform is a unitary isomorphism from L^2(R)
to itself.
Pontryagin duality is the grand generalization of these two
examples! Any locally compact Hausdorff abelian group A has
a "dual" A* consisting of all continuous homomorphisms from A
to S^1. The dual is again a locally compact Hausdorff abelian
group - or "LCA group", for short. When you take duals twice,
you get back where you started. And the Fourier transform gives
a unitary isomorphism between the Hilbert spaces L^2(A) and L^2(A*).
It's fun to take the Pontryagin duals of specific groups, or
specific classes of groups, and see what we get. We've already
seen that the dual of S^1 is Z, the dual of Z is S^1, and the
dual of R is R. More generally the dual of the n-dimensional
torus is Z^n, and vice versa, while the dual of R^n is isomorphic
to R^n. What can we glean from these examples?
Well, any discrete abelian group is an LCA group - a good example
is Z^n. So is any compact Hausdorff abelian group - a good example
is the n-dimensional torus. And there's a nice general theorem
saying that the dual of any group of the first kind is a group of
the second kind, and vice versa!
In particular, if we have an abelian group that's both compact
and discrete, its dual must be too. But the only abelian
groups like this are the *finite* abelian groups - products of
finite cyclic groups Z/n. So, this collection of groups is closed
under Pontryagin duality!
In fact, it's easy to see that for any finite abelian group,
A* is isomorphic to A. But not canonically! To get a canonical
isomorphism we need to take duals twice: for any LCA group,
we get a canonical isomorphism between A and A**. This should
remind you of duality for finite-dimensional vector spaces -
another famous collection of LCA groups that's closed under
Pontryagin duality.
You can take any collection of LCA groups, stare at it through
the looking-glass of Pontryagin duality, and see what it looks
like. I've mentioned a few examples so far:
A is compact iff A* is discrete.
A is finite iff A* is finite.
A is a finite-dimensional vector space iff A*
is a finite-dimensional vector space.
Here are some fancier ones:
A is torsion-free and discrete iff A* is connected and compact.
A is compact and metrizable iff A* is countable.
A is a Lie group iff A* has finite rank.
A is metrizable iff A* is sigma-compact.
A is second countable iff A* is second countable.
If you know more snappy results like this, tell me! I'm
collecting them - they're sort of addictive.
Because Pontryagin duality turns compact LCA groups into discrete
ones - and vice versa - we can use it to turn some topology questions
into algebra questions, and vice versa. After all, a discrete
abelian group has no more structure than an "abstract" abelian
group - one without a topology!
Sometimes this change of viewpoint helps, but sometimes it merely
reveals how hard a problem really is.
For example, here's an innocent-sounding question: what are the
compact path-connected LCA groups? The obvious example is
the circle. More generally, we could take any product of
circles - even an *infinite* product. Are there any others?
It turns out that this question cannot be settled by
Zermelo-Fraenkel set theory together with the axiom of choice!
Here's why. An LCA group is compact and path-connected iff
its dual is a "Whitehead group". What's that? It's an abelian
group A such any short exact sequence of abelian groups like this
splits:
0 -> Z -> B -> A -> 0
where Z is the integers and B is any abelian group.
We call this sort of short exact sequence an "extension of A by Z".
So, if you want to show off your sophistication, you can say that A
is a Whitehead group if "Ext(A,Z) = 0".
The obvious examples of Whitehead groups are free abelian
groups. Indeed, these are precisely the guys whose Pontryagin
dual is a product of circles! So the question is: are there
any others? Or is every Whitehead group a free abelian group?
This is a famous old problem, called the Whitehead problem:
10) Wikipedia, Whitehead problem,
http://en.wikipedia.org/wiki/Whitehead_problem
In 1971, the logician Saharon Shelah showed the answer to this
problem was undecidable using the axioms of ZFC! This was one
of the first problems mathematicians really cared about that
turned out to be undecidable.
If you want an easy introduction to Pontryagin duality and the
structure of LCA groups, you can't beat this:
10) Sidney A. Morris, Pontryagin Duality and the Structure of
Locally Compact Abelian Groups, Cambridge U. Press, Cambridge,
1977.
This classic treatment is still great, too:
14) Lev S. Pontrjagin, Topological Groups, Princeton University
Press, Princeton, 1939.
To dig deeper, you need to read this - it's a real mine of
information:
15) E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I,
Springer, Berlin, 1979.
This book has a lot of interesting newer results:
16) David L. Armacost, The Structure of Locally Compact Abelian
Groups, Dekker, New York, 1981.
In particular, this is where I learned about path-connected
LCA groups and the Whitehead problem.
I'd like to dedicate this issue of This Week's Finds to my father,
Peter Baez, who died yesterday around midnight at the age of 87.
His health had been failing for a long time, so this did not come
as a shock. It's a curious coincidence that I was already writing
an issue about minerals, since my dad majored in chemistry and
returned to school for a master's in soil science after serving
in the Army in World War II. After that he worked in the Blackfeet
Nation in Browning Montana, riding around in a jeep, digging up
soil samples, and testing them back at the lab for the Army Corps
of Engineers. When he found "medicine wheels" - stone circles laid
down by the native Americans for ritual reasons - he would report
them to his friend the archeologist Tom Kehoe. Later he moved to
California, became an editor for the Forest Service, and met my
mother.
He got me interested in science at an early age because he was always
taking me to museums, bringing me books from the public library, and
so on. As a little kid, when I spilled something, he'd say "So you
don't believe in the law of gravity?" He liked to joke around.
Whenever I said an ungrammatical sentence, he'd tease me for it.
"I'm not that hungry." "What do you mean? You're not *how* hungry?"
I learned a lot of math, physics and chemistry from his 1947 edition
of the CRC Handbook of Chemistry and Physics - an edition so old that
it listed "mesothorium" among the radioactive isotopes. He brought
home the book "From Frege to Goedel" - a sourcebook in mathematical
logic - because it was in the math section of the library and he
misread "Goedel" as "Googol". He knew I liked large numbers! I
didn't understand much of it, but it had a big effect on me.
I owe a lot to him.
-----------------------------------------------------------------------
Quote of the Week:
People like us, who believe in physics, know that the distinction
between past, present and future is only a stubbornly persistent
illusion. - Albert Einstein
-----------------------------------------------------------------------
Addenda: I thank Michael Barr, Kevin Buzzard and Mike Stay for some
interesting comments. Mike Stay pointed out an interesting book on how
humans may affect the future of mineral evolution:
17) Jan Zalasiewicz, The Earth After Us: What Legacy Will Humans Leave
in the Rocks?, Oxford University Press, Oxford, 2009.
It's not 2009 yet, but the best books about the future are actually
published in the future. Here's a quote:
The surface of the Earth is no place to preserve deep history.
This is in spite of - and in large part because of - the many
events that have taken place on it. The surface of the future
Earth, one hundred million years now, will not have preserved
evidence of contemporary human activity. One can be quite
categorical about this. Whatever arrangement of oceans and
continents, or whatever state of cool or warmth will exist then,
the Earth's surface will have been wiped clean of human traces.
Thus, one hundred million years from now, nothing will be left of
our contemporary human empire at the Earth's surface. Our planet
is too active, its surface too energetic, too abrasive, too corrosive,
to allow even (say) the Egyptian Pyramids to exist for even a hundredth
of that time. Leave a building carved out of solid diamond - were
it even to be as big as the Ritz - exposed to the elements for that
long and it would be worn away quite inexorably.
So there will be no corroded cities amid the jungle that will,
then, cover most of the land surface, no skyscraper remains akin
to some future Angkor Wat for future archaeologists to pore over.
Structures such as those might survive at the surface for thousands
of years, but not for many millions.
Kevin Buzzard had some interesting comments on Pontryagin duality in
number theory:
I don't know any more general theorems of the form "G has X iff its
dual has Y" but, since lecturing on Tate's thesis, I learnt some
more nice examples of Pontrjagin duals.
As you know well, number theorists like to complete fields with
respect to norms. The rational numbers are too rich to understand
completely, so we choose a norm on them, and then we complete with
respect to that norm, and we get either the reals, or, a couple of
thousand years later, the p-adics. Now a complete field is a much
better gadget to have because there's a chance we'll be able to do
analysis on this field. Indeed, for example, it's possible to set up a
theory of Banach spaces etc for any complete field, and this isn't
just for fun---e.g. Serre showed in the 1960s how to simplify some of
Dwork's work on zeta functions of hypersurfaces using standard
theorems of analysis of Banach spaces, applied to Banach spaces over
the p-adic numbers, and there are oodles of other examples within
number theory (leading up to an entire "p-adic Langlands programme"
nowadays). But ideally, as well as continuity and differentiation etc,
it's nice to be able to do integration as well, and for that you might
need a measure, like Haar measure for example, and if you want to use
Haar measure then you want your complete field to be locally compact
too.
Now, perhaps surprisingly, Weil (I think it was Weil; it might have
been earlier though and perhaps I'm doing someone a disservice)
managed to completely classify normed fields which were both complete
and locally compact. They are: the reals and finite field extensions,
the p-adic numbers and finite field extensions, and the fields
F_p((t)) and finite field extensions. [given a complete normed field,
there's a unique extension of the norm to any finite field extension
and the extension is still complete--for the same sorts of reasons
that there's only one vector space norm on R^n up to equivalence and
it's complete].
Tate in his thesis proves the following result (it's not too hard but
it's crucial for him): if K is a complete locally compact normed field
(considered as a locally compact abelian group under addition), then K
is isomorphic to its own Pontrjagin dual. The isomorphism is
non-canonical because you have to decide where 1 goes, but after
you've made that decision there's a unique natural topological and
algebraic isomorphism. So R and C are their own Pontrjagin
duals---but the p-adic numbers are also self-dual in this way.
One of the reasons (perhaps historically the main reason) that
number theorists are interested in complete fields is that given a
more "global" object, like a number field (a finite extension of the
rationals), one way of understanding it is by understanding its
completions. Before I lectured my class on Tate's thesis, if someone
had asked me to motivate the definition of the adeles, I would have
said that to study a number field k it's easiest to think about it
locally, so we complete with respect to a norm, but we can't choose a
natural norm, so we choose all of them, and then we "multiply them all
together" so we can get to them all at once. This is not really an
ideal answer.
But here's a completely different way of motivating them. Classically,
people interested in automorphic forms would typically consider
functions on a Lie group G which were invariant, or transformed well,
under a well-chosen discrete subgroup: for example one might want to
consider smooth functions on R satisfying f(x)=f(x+1)---a very
interesting class of functions---or perhaps functions on GL_2(R) which
are invariant under GL_2(Z) [and now you're well on the way to
inventing/discovering the theory of modular forms]. In fact Z lives in
R very nicely:
0-->Z-->R-->R/Z-->0
and furthermore there's a bit of magic in this picture: it's
self-dual with respect to Pontrjagin duality!
But Z is awkward. It's not a field. You really begin to see the
awkwardness if you're Hecke in the 1930s trying to figure out what the
correct notion of a Dirichlet character is when working with the
integers not of Q but of a finite extension of Q. The problem is that
the integers in a general number field aren't a principal ideal
domain so it's sometimes hard to get your hands on "local
information": prime ideals aren't in general principal so you can't
always evaluate a global object at one *number* (analogous to the
prime number p) to get local information.
So let's try and fix this up. Let k be a number field. k is definitely
a global object, like Z, and it's also much easier to manage---it's a
field rather than just a ring. The question is: what is the analogue
of
0-->Z-->R-->R/Z-->0
if we replace Z by k, a number field? [let's give k the discrete
topology, because it has no natural topology other than this]. Even
replacing Z by the rationals Q (with the discrete topology) is an
interesting question! "Z is to R as Q is to...what?".
Well, Tate proposes the following: Z is to R as Q is to the adeles of
Q! More generally Z is to R as k (a number field) is to its
adeles. And the argument he could use to justify this isn't
number-theoretic at all, in some sense---it's coming entirely from
Pontrjagin duality! Tate shows that the Pontrjagin dual of a number
field k is A_k/k, where A_k is the adeles of k, and k is embedded
diagonally! Now the analogue of the beautiful self-dual picture
0->Z->R->R/Z->0 is going to be
0-->k-->???-->A_k/k-->0
and the natural candidate for ??? is of course now the adeles
A_k. These gadgets have appeared "magically" in some sense---the
argument seems to me to be topological rather than arithmetic
(although of course there is more than one choice for ??? and perhaps
the argument that the adeles are the right thing to put in the middle
is number-theoretic).
Here's Tate's proof. Pontrjagin duality sends direct sums to direct
products and vice-versa. So neither of them in particularly
"symmetric"---both get changed. But Tate observes that *restricted*
direct products get sent to restricted direct products! Let k be a
number field. Let k_v denote a typical completion of k (so if k=Q then
k_v is either R or Q_p). We know k_v is complete and it's easy to
check it's locally compact (this doesn't use Weil's
classification---it's the easy way around). So k_v is self-dual. So
the restricted product of all the k_v (that is, the adeles), is
locally compact, and dual to the restricted product of all the
(k_v)^*, and (k_v)^* is k_v again, so the adeles of k are self-dual!!
Tate then checks that k (embedded diagonally in A_k) is discrete and
equal to its own annihilator (if you choose all the local isomorphisms
k_v=(k_v)^* just right), and hence by the "Galois correspondence"
between closed subgroups of G and closed subgroups of G^* he deduces
that A_k/k is the dual of k. In particular
0-->k-->A_k-->A_k/k-->0
looks like a very natural analogue of 0-->Z-->R-->R/Z-->0; the
quotient is compact, the sub is discrete, and the diagram is
self-Pont-dual.
Within about 10 years of Tate's thesis it's visibly clear in the
literature that there has been a seismic shift: there seem to be as
many people studying G(Q) \ G(adeles) as there are studying G(Z) \
G(R) in the theory of automorphic forms, and the adelic approach has
the advantage that, although less concrete, it has "truly local"
components, thus motivating the representation theory of p-adic
groups, the Langlands programme, and lots of other things.
Kevin
Michael Barr wrote:
Did you know that there is a *-autonomous category of topological
abelian groups that includes all the LCA groups and whose duality
extends that of Pontrjagin? The groups are characterized by the
property that among all topological groups on the same underlying
abelian group and with the same set of continuous homomorphisms to the
circle, these have the finest topology. It is not obvious that such a
finest exists, but it does and that is the key.
He has a paper on this:
18) Michael Barr, On duality of topological abelian groups,
available at ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/abgp.pdf
For more discussion visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2008/12/this_weeks_finds_in_mathematic_34.html
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