\Ohlllll' U. '\lIlllhlT 4 TIll' ]\l'\\slettlT of thl' l\lathl'l1latical ,\ssol'iation of ,\l1ll'rica \lI!':lIst 11)1)3 See Page 3 Section Distinguished TeachingAwards See Page 6 FOCUS August 1993 Table ofContents FOCUS is published by The Mathemati­ cal Association of America, 1529 Eighteenth Street Northwest, Washing­ ton, DC 20036-1385, six times a year: February, April, June, August, October, Features and December. Editor: Keith 1. Devlin, St. Mary's Col­ Fermat's Last Theorem 3 lege ofCalifornia Associate Editor: Donald J. Albers, MAAAssociate Executive Director, and Director ofPublications and Programs Twenty-Second Chair ofthe MAA Newsletter Edito­ Mathematical Olympiad 5 rial Committee: Joseph A. Gallian, University of Minnesota at Duluth SectionAwards for Distinguished Managing Editor: Harry Waldman, MAA Collegeor University Mathematics Production Specialist: Amy E. Teaching 6 Stephenson, MAA Subscribers should address letters to the editor: Keith Devlin, St. Mary's College NSF Conference on Statistics Education 8 of California, Moraga, CA 94575. E-mail: [email protected] The FOCUS subscription price to indi­ vidual members of the Association is The 1992MAAAnnualReport 13 $6.00, included in the annual dues. (An­ nual dues for regular members, exclusive of annual subscription prices for MAA journals, are $68.00. Student and unem­ Starting an Industrial Postdoctoral Program 32 ployed members receive a 66 percent discount; emeritus members receive a 50 percent discount; new members receive a 40 percent discount for the first two membership years.) Copyright © 1993 by The Mathematical Association of America (Incorporated). Educational institutions may reproduce articles for their own use, but not for sale, provided that the following citation is Departments used: "Reprinted with permission of FOCUS, the Newsletter ofThe Math­ ematical Association ofAmerica Editorial 10 (Incorporated), 1993." Second-class postage paid at Washing­ ton, DC and additional mailing offices. Personal Opinion 11 Postmaster: Send address changes to the Membership and Subscriptions Depart­ ment, The Mathematical Association of America, 1529 Eighteenth Street North­ Employment Opportunities 35 west, Washington, DC 20036-1385. ISSN: 0731-2040 Printed in the United States of America. Printed on recycled paper August1993 FOCUS i = x(x - an)(x+if) Fermat's Last Theorem, in the way proposed by Taniyama. Follow­ ing an appropriate re-formulation by Jean-Pierre Serrein Paris, Kenneth Ribet in a Theorem at Last Berkeley strengthened Frey's original con­ cept to the point where it was possible to Keith Devlin, Fernando Gouvea,andAndrew Granville provethat the existenceofacounterexample to the LastTheoremwould lead to the exist­ After defying all attempts at a solution for different direction, and rests on an amazing ence ofan elliptic curve which could not be 350 years, Fermat's Last Theorem finally connection, established during the last de- modular, and hence would contradict the tookits placeamongthe knowntheoremsof cade, between the Last Theorem and the Shimura-Taniyama-Weil conjecture. mathematics in June ofthis year. theory ofelliptic curves, that is, curves de­ termined by equations ofthe form This is the point where Wiles entered the On June 23, during the third of a series of picture.Usingand developingpowerfulnew lectures at a conference held at the Newton i=r +ax+b, methodsofBarryMazur(Harvard),Matthias Institute in Cambridge, British mathemati­ where a and b are integers. Flach (Heidelberg), Victor Kolyvagin cian Dr Andrew Wiles, of Princeton (Steklov Institute), and others, Wiles even­ University, sketched a proof the Shimura­ The path that led to the June 23 announce­ tually succeeded in establishing the Taniyama-Weil conjecture for semi-stable ment began in 1955, when the Japanese Shimura-Taniyama-Weil conjecture for an elliptic curves. As Kenneth Ribet, of the mathematician Yutaka Taniyama proposed importantclass ofellipticcurves(thosewith UniversityofCalifomiaatBerkeley, showed that there should be a connection between square-free "conductors"), which includes someyears ago, Fermat's LastTheoremis a elliptic curves and another well-understood those relevant to proving Fermat's Last corollary ofthis result. class of curves, known as modular curves. Oneshouldbe able to establish a connection Theorem. In the years since Fermat made his famous betweenany given elliptic curveand amodu­ For the Cambridge conference, Wiles had marginal note in his copy of Diophantus' lar curve, and this connection would announcedhis lectures, a series ofthree given Arithmetica, asserting, without proof, that "control" many ofthe properties ofthe ini­ on successive days, with the highly unspe­ equations of the form tial curve. cific title "ModularForms, Elliptic Curves, x"+y"=z" Taniyama'sconjecture was made morepre­ and Galois Representations." Prior to his lectures, he refused to give a hint as to what have no integer solutions for values of the cise in 1968 by Andre Weil, who showed they mightcontain.Eventhough, by the third exponent n greater than 2, many mathema­ how to determine the exact modular curve talk, many in the audience had guessed he ticians, professionals and amateurs alike, that should be connected to a given elliptic mighthave crackedFermat'sLastTheorem, have tried to find a proof. Every few years, curve. few dreamt that he could have proved so the newspapersreportyet anotherpurported In 1971 the first significant evidence in fa­ much more, and there was an audible gasp solution, which is subsequently found to be vor of this abstract understanding of as he wrote the final result on the black­ lacking in some way. equations was given by Goro Shimura, a board. To understand why Wiles has succeeded Japanese mathematician at Princeton Uni­ Given the history of attempts to prove the where so many before himhave failed, one versity, who showed that it works for a very Last Theorem, readers will doubtless view needs to know something ofthe recent his­ special class of equations. As a result, this latest announcement with some initial tory ofthe problem. Taniyama's proposal eventually became knownas the Shimura-Taniyama-Weil con­ skepticism. But it should be stressed that To number theorists, until the last decade jecture. Wiles' work is not a chain of reasoning as the question seemed completely unassail­ strong as its weakest link. Instead it is a able, despite a number of significant Additional evidence in support of the con­ bedrock ofideas, solid and rigid, a rich and advances that had been made as a result of jecture came from the fact that its nature profound theory that will hold up even if a attempts to solve it, among them the 19th allowed for a substantial amountofnumeri­ few details need altering. Considering the century work of Kummer on cyclotomic cal testing by computer: all curves that were enormouscomplexityofthis workit will, of fields and ideal theory. examined seemed to be modular. course, take time to be absolutely certain that there are no hidden flaws, but the ex­ Morerecently, there was GerdFaltings, 1983 But so far, no one knew of any connection perts feel confident that even any necessary proofofthe Mordell Conjecture, which im­ between this very abstract conjecture and changes will be possible. Many of these plied that for each exponent, the Fermat Fermat'sLastTheorem.Thingschangeddra­ expertswere attendingWiles'talks, and from equationcouldhaveat mosta finite number maticallyin 1986, whenGerhardFrey, from what he said and the way he said it, they of solutions. But, despite much simpler Saarbrucken, discovered a most surprising were satisfied that the techniques are, in es­ proofs ofVojta and Bombieri, it seems un­ and innovative link between the two. sence, sound. likely that such methods can be modified to Whathe realized was thatifen = an +if,then show that there really are no solutions. it seemedunlikelythat one couldunderstand The situation is a bit like building a bridge across a chasm. Most attempts to solve the Wiles' approach comes from a somewhat the elliptic curve given by the equation Please see Fermat on page 4 FOCUS August1993 Fermatfrom page 3 Last Theorem made by amateurs are like a The Technical Details single span consisting ofmany short pieces Shortly after Wilesfinished his lecture, DrKenneth Ribet, ofthe University ofCalifornia ofwire tied together. One small flaw and the at Berkeley, who was in the audience. sent out the following summary ofthe proofto entire structure collapses. Wiles' solution is colleagues at Berkeley, where the excitement it createdled to its rapidappearance on the much more like a solid concrete and steel Internet. The message is reproduced here with his permission. bridge, erected on firm foundations laid down by many others. Itmay contain many I imagine that many ofyou have heard rumors about Wiles's announcement a few hours small cracks, but the bridgeitselfstill stands, ago that he can prove Taniyama's conjecture for semistable elliptic curves over Q.This and the road crew can comealonglater to fill caseofthe Taniyamaconjecture implies Fermat'sLastTheorem, in view ofthe result that in the cracks. Of course, even solidly built I proved a few years ago. (I proved that the "Frey elliptic curve" constructed from a bridges do fall down occasionally, but the possible solutionto Fermat'sequationcannotbe modular, i.e., satisfy Taniyama'sConjec­ natureof Wiles ' work, whichinvolvesmany ture. On the other hand, it is easy to see that it is semistable.) appreciable, profound new ideas,