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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 (9121031), 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 arRahman 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 arRahman 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 linebyline 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 (11701250) 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 (9401003) 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 mid12th 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://wwwgroups.dcs.stand.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 secondhand 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 singlehandedly 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 Syriacspeaking scholars of Greek in the Persian city of Jundishapur, and Arabic and Pahlavispeaking Muslim scholars of Syriac, including the Nestorian Hunayn Ibn Ishak (809873) 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), 177219. 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 farreaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms. To motivate it, recall the classical KroneckerWeber 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 padic 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 padic 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 KroneckerWeber theorem. One can obtain complete information about the maximal abelian quotient of a group by considering its onedimensional representations. The above statement of the abelian class field theory may then be reformulated as saying that onedimensional representations of Gal(Fbar/F) are essentially in bijection with onedimensional 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 *ndimensional 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 ndimensional 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 finitedimensional representations, equipped with the structure of the tensor product. Therefore looking at ndimensional 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 twodimensional 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 ndimensional 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 qexpansion 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 (n1)storder Taylor series padic integers Taylor series padic numbers Laurent series Adeles for the rationals Adeles for the rational functions Fields Onepoint spaces Homomorphisms to fields Maps from onepoint 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 http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumpingoff point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html 


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