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

  1. Oct 12, 2006 #1
    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.

    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

    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 [Broken]

    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

    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?,

    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,

    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

    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 [Broken]

    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


    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




    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


    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":


    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

    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

    Last edited by a moderator: May 2, 2017
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