This Week's Finds in Mathematical Physics (Week 221)

John Baez

Also available as http://math.ucr.edu/home/baez/week221.html

September 18, 2005
This Week's Finds in Mathematical Physics - Week 221
John Baez

After going to the Streetfest this summer, I wandered around China.
I began by going to a big conference in Beijing, the 22nd
International Congress on the History of Science. I learned some
interesting stuff. For example:

The eleventh century was the golden age of Andalusian astronomy
and mathematics, with a lot of innovation in astrolabes. During
the Caliphate (912-1031), three quarters of all mathematical
manuscripts were produced in Cordoba, most of the rest in
Sevilla, and only a few in Granada in Toledo.

<P>
I didn't understand the mathematical predominance of Cordoba when
I first heard about it, but the underlying reason is simple.
The first great Muslim dynasty were the Ummayads, who ruled from
Damascus. They were massacred by the Abbasids in 750, who then
moved the capital to Baghdad. When Abd ar-Rahman fled Damascus
in 750 as the only Ummayyad survivor of this massacre, he went
to Spain, which had already been invaded by Muslim Berbers in 711.

Abd ar-Rahman made Cordoba his capital. And, by enforcing a certain
level of religious tolerance, he made this city into *the place to
be* for Muslims, Jews and Christians - the "ornament of the world",
and a beacon of learning - until it was sacked by Berber troops in
1009.

Other cities in Andalusia became important later. The great
philosopher Ibn Rushd - known to Westerners by the Latin name
"Averroes" - was born in Cordoba in 1128. He later became a judge
there. He studied mathematics, medicine, and astronomy, and wrote
detailed line-by-line commentaries on the works of Aristotle. It
was through these commentaries that most of Aristotle's works,
including his Physics, found their way into Western Europe! By 1177,
the bishop of Paris had banned the teaching of many of these new
ideas - but to little effect.

Toledo seems to have only gained real prominence after Alfonso VI
made it his capital upon capturing it in 1085 as part of the
Christian "reconquista". By the 1200s, it became a lively center
for translating Arabic and Hebrew texts into Latin.

Mathematics also passed from the Arabs to Western Europe in other
ways. Fibonacci (1170-1250) studied Arabic accounting methods in
North Africa where his father was a diplomat. His book Liber Abaci
was important in transmitting the Indian system of numerals
(including zero) from the Arabs to Europe. However, he wasn't the
first to bring these numbers to Europe. They'd been around for over
200 years!

For example: Gerbert d'Aurillac (940-1003) spent years studying
mathematics in various Andalusian cities including Cordoba. On
his return to France, he wrote a book about a cumbersome sort of
"abacus" labelled by a Western form of Arabic numerals. This
remained popular in intellectual circles until the mid-12th century.

Amusingly, Arabic numerals were also called "dust numerals" since
they were used in calculations on an easily erasable "dust board".
Their use was described in the Liber Pulveris, or "book of dust".

I want to learn more about Andalusian science! I found this book
a great place to start - it's really fascinating:

1) Maria Rose Menocal, The Ornament of the World: How Muslims, Jews
and Christians Created a Culture of Tolerance in Medieval Spain,
Little, Brown and Co., 2002.

For something quick and pretty, try this:

2) Steve Edwards, Tilings from the Alhambra,
http://www2.spsu.edu/math/tile/grammar/moor.htm

Apparently 13 of the 17 planar symmetry groups can be found in tile
patterns in the Alhambra, a Moorish palace built in Granada in the
1300s.

If you want to dig deeper, you can try the references here:

3) J. J. O'Connor and E. F. Robertson, Arabic mathematics:
forgotten brilliance?,
http://www-groups.dcs.st-and.ac.uk/~...thematics.html

For more on Fibonacci and Arabic mathematics, try this paper by
Charles Burnett, who spoke on this subject in Beijing:

4) Charles Burnett, Leonard of Pisa and Arabic Arithmetic,
http://muslimheritage.com/topics/def...?ArticleID=472

Another interesting talk in Beijing was about the role of the
Syriac language in the transmission of Greek science to Europe.
Many important texts didn't get translated directly from Greek to
Arabic! Instead, they were first translated into *Syriac*.

I don't understand the details yet, but luckily there's a great
book on the subject, available free online:

5) De Lacy O'Leary, How Greek Science Passed to the Arabs,
Routledge & Kegan Paul Ltd, 1949. Also available at
http://www.aina.org/books/hgsptta.htm

So, medieval Europe learned a lot of Greek science by reading Latin
translations of Arab translations of Syriac translations of
second-hand copies of the original Greek texts!

I want to read this book, too:

6) Scott L. Montgomery, Science in Translation: Movements of
Knowledge through Cultures and Time, U. of Chicago Press, 2000.
Review by William R. Everdell available at MAA Online,
http://www.maa.org/reviews/scitrans.html

The historian of science John Stachel, famous for his studies of
Einstein, says this book "strikes a blow at one of the founding
myths of 'Western Civilization'" - namely, that Renaissance Europeans
single-handedly picked up doing science where the Greeks left off.
As Everdell writes in his review:

Perhaps the best of the book's many delightful challenges
to conventional wisdom comes in the first section on the
translations of Greek science. Here we learn why it is
ridiculous to use a phrase like "the Renaissance recovery
of the Greek classics"; that in fact the Renaissance recovered
very little from the original Greek and that it was long before
the Renaissance that Aristotle and Ptolemy, to name the two most
important examples, were finally translated into Latin. What
the Renaissance did was to create a myth by eliminating all the
intermediate steps in the transmission. To assume that Greek
was translated into Arabic "still essentially erases centuries
of history" (p. 93). What was translated into Arabic was
usually Syriac, and the translators were neither Arabs (as
the great Muslim historian Ibn Khaldun admitted) nor Muslims.
The real story involves Sanskrit compilers of ancient Babylonian
astronomy, Nestorian Christian Syriac-speaking scholars of
Greek in the Persian city of Jundishapur, and Arabic- and
Pahlavi-speaking Muslim scholars of Syriac, including the
Nestorian Hunayn Ibn Ishak (809-873) of Baghdad, "the greatest
of all translators during this era" (p. 98).

And now for something completely different: the Langlands program!
I want to keep going on my gradual quest to understand and explain
this profoundly difficult hunk of mathematics, which connects
number theory to representations of algebraic groups. I've found
this introduction to be really helpful:

7) Stephen Gelbart: An elementary introduction to the Langlands
program, Bulletin of the AMS 10 (1984), 177-219.

There are a lot of more detailed sources of information on the
Langlands program, but the problem for the beginner (me) is that
the overall goal gets swamped in a mass of technicalities.
Gelbart's introduction does the best at avoiding this problem.

I've also found parts of this article to be helpful:

8) Edward Frenkel, Recent advances in the Langlands program, available
at math.AG/0303074.

It focuses on the "geometric Langlands program", which I'd rather
not talk about now. But, it starts with a pretty clear introduction
to the basic Langlands stuff... at least, clear to me after I've
battered my head on this for about a year!

If you know some number theory or you've followed recent issues of
This Week's Finds (especially "week217" and "week218") it should make
sense, so I'll quote it:

The Langlands Program has emerged in the late 60's in the form of
a series of far-reaching conjectures tying together seemingly
unrelated objects in number theory, algebraic geometry, and the
theory of automorphic forms. To motivate it, recall the classical
Kronecker-Weber theorem which describes the maximal abelian extension
Q^{ab} of the field Q of rational numbers (i.e., the maximal extension
of Q whose Galois group is abelian). This theorem states that Q^{ab}
is obtained by adjoining to Q all roots of unity; in other words,
Q^{ab} is the union of all cyclotomic fields Q(1^{1/N}) obtained
by adjoining to Q a primitive Nth root of unity

1^{1/N}

The Galois group Gal(Q(1^{1/N})/Q) of automorphisms of Q(1^{1/N})
preserving Q is isomorphic to the group (Z/N)* of units of the
ring Z/N. Indeed, each element m in (Z/N)*, viewed as an integer
relatively prime to N, gives rise to an automorphism of Q(1^{1/N})
which sends

1^{1/N}

to

1^{m/N}.

Therefore we obtain that the Galois group Gal(Q^{ab}/Q), or,
equivalently, the maximal abelian quotient of Gal(Qbar/Q),
where Qbar is an algebraic closure of Q, is isomorphic to the
projective limit of the groups (Z/N)* with respect to the system
of surjections

(Z/N)* -> (Z/M)*

for M dividing N. This projective limit is nothing but the direct
product of the multiplicative groups of the rings of p-adic
integers, Z_p*, where p runs over the set of all primes. Thus,
we obtain that

Gal(Q^{ab}/Q) = product_p Z_p*.

The abelian class field theory gives a similar description for the
maximal abelian quotient Gal(F^ab/F) of the Galois group Gal(Fbar/F),
where F is an arbitrary global field, i.e., a finite extension of
Q (number field), or the field of rational functions on a smooth
projective curve defined over a finite field (function field).
Namely, Gal(F^ab/F) is almost isomorphic to the quotient A(F)*/F*,
where A(F) is the ring of adeles of F, a subring in the direct
product of all completions of F. Here we use the word "almost"
because we need to take the group of components of this quotient
if F is a number field, or its profinite completion if F is a
function field.

When F = Q the ring A(Q) is a subring of the direct product of the
fields Q_p of p-adic numbers and the field R of real numbers, and
the quotient A(F)*/F* is isomorphic to

R+ x product_p Z*_p.

where R+ is the multiplicative group of positive real numbers.
Hence the group of its components is

product_p Z*_p

in agreement with the Kronecker-Weber theorem.

One can obtain complete information about the maximal abelian
quotient of a group by considering its one-dimensional
representations. The above statement of the abelian class field
theory may then be reformulated as saying that one-dimensional
representations of Gal(Fbar/F) are essentially in bijection with
one-dimensional representations of the abelian group

A(F)* = GL(1,A(F))

which occur in the space of functions on

A(F)*/F* = GL(1,A(F))/GL(1,F)

A marvelous insight of Robert Langlands was to conjecture that
there exists a similar description of *n-dimensional
representations* of Gal(Fbar/F). Namely, he proposed that those
may be related to irreducible representations of the group
GL(n,A(F)) which are *automorphic*, that is those occurring in
the space of functions on the quotient

GL(n,A(F))/GL(n,F)

This relation is now called the *Langlands correspondence*.

At this point one might ask a legitimate question: why is it
important to know what the n-dimensional representations of the
Galois group look like, and why is it useful to relate them to
things like automorphic representations? There are indeed many
reasons for that. First of all, it should be remarked that
according to the Tannakian philosophy, one can reconstruct a
group from the category of its finite-dimensional representations,
equipped with the structure of the tensor product. Therefore
looking at n-dimensional representations of the Galois group is
a natural step towards understanding its structure. But even
more importantly, one finds many interesting representations of
Galois groups in "nature".

For example, the group Gal(Qbar/Q) will act on the geometric
invariants (such as the etale cohomologies) of an algebraic variety
defined over Q. Thus, if we take an elliptic curve E over Q,
then we will obtain a two-dimensional Galois representation on its
first etale cohomology. This representation contains a lot of
important information about the curve E, such as the number of
points of E over Z/p for various primes p.

The point is that the Langlands correspondence is supposed to
relate n-dimensional Galois representations to automorphic
representations of GL(n,A(F)) in such a way that the data on
the Galois side, such as the number of points of E over Z/p,
are translated into something more tractable on the automorphic
side, such as the coefficients in the q-expansion of the modular
forms that encapsulate automorphic representations of GL(2,A(Q)).

More precisely, one asks that under the Langlands correspondence
certain natural invariants attached to the Galois representations
and to the automorphic representations be matched. These
invariants are the *Frobenius conjugacy classes* on the Galois
side and the *Hecke eigenvalues* on the automorphic side.

Since I haven't talked about Hecke operators yet, I'll stop here!

But, someday I should really explain the ideas behind the baby
"abelian" case of the Langlands philosophy in simpler terms than
Frenkel does here. The abelian case goes back way before Langlands:
it's called "class field theory". And, it's all about exploiting
this analogy, which I last mentioned in "week218":

NUMBER THEORY COMPLEX GEOMETRY

Integers Polynomial functions on the complex plane
Rational numbers Rational functions on the complex plane
Prime numbers Points in the complex plane
Integers mod p^n (n-1)st-order Taylor series
p-adic integers Taylor series
p-adic numbers Laurent series
Adeles for the rationals Adeles for the rational functions
Fields One-point spaces
Homomorphisms to fields Maps from one-point spaces
Algebraic number fields Branched covering spaces of the complex plane

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