art - mathematics, monsters & manifolds

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Mathematics: Monsters and ManifoldsAuthor(s): Rick NorwoodSource: The Wilson Quarterly (1976-), Vol. 6, No. 2 (Spring, 1982), pp. 98-111Published by: Wilson QuarterlyStable URL: http://www.jstor.org/stable/40256268 .

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Mathematics:

MONSTERS N D

M NIFOLDS

IzaakWalton once likened mathematics tofishing:Bothare sol-itary pursuits that "can never befullylearnt." Yet mathematicshas a loathsomereputation. It frightensmany peopleand maybore the rest.In its highest reaches, it seemsto defycommonsense. There are17,000mathematicians in the United Statestoday, almost all of thememployed byuniversitiesor by flour-ishing "high-tech"corporations. Their ranks include brillianttheorists, yeoman problem-solvers,colorful eccentrics, class-roomdrones. Thedisciplinehas progressedby leapsand boundssince 1900;the past decade has beenespeciallyfruitful.But fewAmericans understand the achievementsof modern mathe-matics orits contributions to theworkingsof anindustrial soci-ety. The federal governmentis moreappreciative;Washingtonplans to create twonational mathematical"institutes," one atthe Universityof Minnesota,the otherat the Universityof Cali-fornia, Berkeley.Here, mathematician Rick Norwoodexplainssome ofthe advances of recentyears and describes theun-solvedproblemsthat his colleaguesseekto answer.

Smith Collection.Rare Books andManuscriptsLibrary.ColumbiaUniversity.

byRickNorwood

All mathematics is dividedinto three parts.Roughly,theseparts are the studyof numbersystems,calledalgebra;the study of geometricalspaces,calledtopology;and the study of functions,called analysis.Thereare alsoa fewislands- number theoryand set

theory, for example-

and two vastcontinents thathave broken off fromthe mainland and aredriftingout to sea:computerscience andstatistics.

Mostmathematiciansidentifywith one ormore of

The WilsonQuarterlySpring198298







MATHEMATICS

thesedisciplines,and it is almostimpossibleo keepup withanybut the mostimportantresults outsideone's ownarea,a songof woe heardin otherlandsthanmathematics.

The totalamountofmathematicswrittenin thiscenturyprobablyexceeds the total ofall previouscenturies takentogether.In 1980,more than64,000pagesofnewmathematicswerepublished.Notevery-one in the field today (perhapsnot anyone)is aNewtonor aGauss,but muchof themathematicsnowbeingdonemeetsextremelyhighstandardsoforigi-nalityandexcellence.Muchhas been discovered hatisstartlingandbeautiful.

Mostpeoplethinkof mathematicsas a dry,cob-webbedscience,writtenbyEuclidontabletsof stoneand passeddownthroughthe agesfromteachertostudent untilall are bent under die load as the do-mainofhoaryeccentricsat oneextreme,and of fear-someschoolmasters t the other.It haslongbeenthebutt of jokes."I have hardly ever met a mathe-maticiancapableof reasoning,"Plato once said.Toseemathematicsexhibit

graceand

elegance,then,is

as surprisingas thesight,in Shogun,of the warriorToranaga ancingonthebattlements.Nevertheless,tdances.

Topologys myownfield. Tome,the fascinationof multidimensionalgeometric shapesis unending.Onecan easilyvisualizeonlythe simplestof thesethings- the curious Kleinbottle,for instance;thesphere; hedoughnut-shapedorus. The rest one canneverreally"see";theycanbe manipulatedonlyinone's head.Or one cantry (asI havetried)toconveytheirpropertiesby employingseven colors of chalk

whilesimultaneouslyusing bodilygesturesto simu-late motion. I'llsavetopologyfor last.First,a briefwordaboutcomputercienceand statistics.

Statistics isthe science thatnearlygotsaccharinbanned n 1977on the basis ofa studyof630cancerpatientsbytheNationalCancer nstituteof Canada. tis alsothe scienceused in the 1979U.S. NationalCancerInstitutestudyof more than3,000patientsthatwon saccharina reprieve.Whenproperlyapplied(e.g., bya life insurancecompany),statistics is as"true"andas useful asanyother mathematics.

Computerscience, meanwhile,has moved intoour livingrooms and seems there tostay. Already,computersbuild Toyotasand play gameswith us.Tomorrow,hey maydo housework.Thedayafter to-morrow,heymaytell us thattheydon'tdo windows.

From (u.ulu.iU- IfMsin MathcmatKs/n John Sulhvvll

< IWOhxSpntwr-YctLiK \<'u York.

TheWilsonQuarterlySpring198299







MATHEMATICS

Hvaristc Galois( 1811 32)

( nun, m mHunLs-

Computershave revolutionizedcommerce,communi-cations,government; theyare an indispensabletoolinengineeringand in all of thephysicalsciences.

While mathematics gave birth to computer sci-ence and statistics(in the case ofthe former,electricalengineeringwas thefather),both havelargelypackedup and moved out, perhaps because of mathe-maticians' schoolmarmish insistenceon right an-swers.

The problems with statistics are obvious.Given

the real-world constraintsoftime,money,or a client'sneeds,there is a tendencyto cut corners. Muchof thestatistical datapublishedthesedays is quite mislead-ing. Computerscientists,for their part, are primarilyinterested in programs that will be reasonablyreli-able. But noprogram is perfect, and a good mathe-maticiancan usuallycomeup with some"input" thatwill overload acomputer'smemory.I can sometimesbefuddlemyownpocketcalculatorby askingit toper-form certain choresthat couldquite easilybe workedout withpenciland paper.

Does it sound asif I take a dim view ofmathe-matics' glamorous offspring?I don't, really. I am im-pressed by the power of modern statistics andstaggered bythe advancesin computers.Ifanything,Iam a littleenvious.I wishsimplyto note a difference.Appliedmathematicsasks, "Howcan weaccomplishsuch-and-suchin a practical and efficientway?"Puremathematicsasks, "What is? What differentkinds ofmathematicalobjectscan exist?"

Take"group theory."Algebra, as I have mentioned, is a study of

number systems. The algebra one learns in highschool is animportant example,but it bearsthe samerelation to what a mathematician meansby algebraasthe study of lions bears tozoology.Mathematiciansclassifynumbersystems accordingto theirproperties,just as zoologists classifyanimals. A warm-bloodedanimal thatsucklesits youngis a mammal. Aset withone operation that is associative and hasidentityandinversesis agroup.(Anexampleof anoperationwouldbe addition or multiplication.)The associativelaw is(a+b)+c = a+(fe+c). In other words, when adding

Rick Norwood,39, is visiting assistant professorofmathematicsat Lehigh Universityand recentlycom-pleteda year at the Institute for AdvancedStudy inPrinceton,N.J. He holds a B.A.(1966)and a Ph.D.(1979)fromthe UniversityofSouthwest Louisiana.

TheWilsonQuarterly/Spring 1982100







MATHEMATICS

three or morenumbers,youcan drop the parentheseswithoutbeingambiguous.The lawofidentityis that0+ a=a + O= a. Thelaw ofinversesis thata + (-a) =(-a) + a = 0.

The wholenumbersform a group,but so do ma-trices, polynomials, and functions. Quite a lot ofthings can play the role of numbers in abstractalgebra,just as, in a courtoflaw, the InteriorDepart-ment, the Sierra Club,and Exxonmay all be consid-ered, legally, to be "individuals." If you find this

troublesome,refer toWhitehead(below).

"Now,it cannotbe tooclearlyunderstoodthat, in sci-ence,technicalterms are namesarbitrarily assigned,likeChristiannamesto children.There canbe noques-tion of the namesbeing rightor wrong.Theymay bejudiciousor injudicious The essentialprinciplein-volvedwasquite clearlyenunciatedin WonderlandtoAliceby HumptyDumpty,whenhe toldher, aproposof his useof words,'I pay them extra and makethemmeanwhat I like.'"

-Alfred NorthWhitehead,AnIntroductiontoMathematics(1911)

Groupsoccurin allbranchesofmathematics,andin theoreticalphysicsas well,notably in the study ofcrystals and quarks. Many important mathematicalproblemscomedown to a questionabout the groupsinvolved.For example, in the theory of "knots" (abranchoftopology),weknowthat foreveryknotthereis agroup.If we canshowthat the groupin questionisthe group of wholenumbers, then we knowthat the

"knot" can be pulled and stretchedinto a

circle;that

the "knot" was reallyuntied to beginwith.For 150years, since the days of Augustin-Louis

Cauchy and Évariste Galois, mathematicians havebeen trying to answer these questions:"What differ-ent typesof groupscan there be? What are they like?Howcan we tellonefromanother?"This is calledtheclassificationproblem.1

Let us concentrateon the finite groups, groupswith a finite number of elements.The smallestfinitegroupconsistsonlyofthe number0 andthe operation+ . It is a group with only one number; in math ter-

minology,a groupoforder one.Onceyouknowthat 0+ 0 = 0,youknoweverythingyouneedto knowaboutthis group.

Thenext smallestgrouphas order two.Oneway

•The term "grouptheory" itself can betracedbackto a letterwritten in 1832 byGalois.The letterwascomposedon the eveof the duel in whichthe20-year-oldGaloiswas killed. Formoreinformation aboutCauchy, Galois, andother

prominent fig-ures,see EricTempleBell's classic MenofMathematics (Simon&Schuster, 1937).

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101







MATHEMATICS

2Thus far, I havecalled the group op-eration "addition"and written it " +,"and I willcontinue todoso,but I couldjustas wellhavecalled it"multiplication" oranythingelse. What'sin a name?Therows,by any other name,wouldadd assweet.

to describe it isby usingthe numbersT (forTrue)andF (for False).When we "add" twonumbers, we areasking for the truth value of thestatement, "Thesenumbersare the same." SoT + T = T,F + F = T, butT + F = F and sodoesF + T. Theidentityin this caseis T, because T added to any number gives thatnumber as an answer.Each number is also its owninverse,because a number added to itselfgives theidentity T. Another way of lookingat the group oforder two isby substituting "even" for T and "odd"

for F. You might think this was a secondgroup oforder two,but mathematicians do not look atit thatway. To a mathematician, the group of T and F is"isomorphic"to the group of evenand odd, becausethe addition rulesare the same for bothgroups.So wesay that there isonlyonegroupof ordertwo.2

There is alsoonly one group of order three (i.e.,with three elements), but there are two differentgroupsof orderfour,and from then on there are oftenmany groupsof agivenorder.

Bya techniqueknown as thecompositionseries,finite groups- groups with a finite number ofele-ments- can alwaysbe brokenup into smallergroupscalled simple groups, somewhat as a wholenumberlike 15 canbe broken into itsprime factors,3 and 5.Thus, mathematicians have beenchiefly concernedwith classifyingthe finitesimple groups.Andin 1980,the processof classificationwascompleted.

The easiest examplesof finitesimple groupsarethe cyclicgroupsofprimeorder (e.g.,a groupof orderthree,since3 is aprime,divisibleonly byitself and1),and the so-calledalternating groupsof order 60 ormore. These are groups one would study in anundergraduate course in abstract algebra. There areinfinitely manyof them,but theyare completelyclas-sified andwellunderstood.

Next,if you pursuedabstract algebraat the grad-uate level,you wouldencounter finitesimplegroupsof the Lie type,named forNorwegianmathematicianMarius Sophus Lie (pronouncedLee).These are noteasy to understand. Even so, finite simple groupsofthe Lietypeare alsocompletelyclassified.

Finite simplegroupsthat are not ofprime order,or alternating,or Lie,are calledsporadic.Theyare thesports, the strange ones. They were the last type offinite

simplegroupto be

classified.We now know thatthere are exactly26 ofthem.Some ofthem have pet names like Monster and

BabyMonster;these two, in fact, were thelast to be








MATHEMATICS

discovered. The order of Monster is808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. In other words, it has that many elements. Itsdiscovery byBernd Fischer of the University ofBielefeld inWestGermanywas one of the mostimpor-tant achievementsin pure math in recentyears.

Whyhave I led you over suchmurky terrain? Torepeat: Groupsare buildingblocks,like the elementsof the periodictable. Some elements- hydrogen,say- are rather basic;we knowpretty well whattheycanbe usedfor, and we use them all the time. Other ele-ments- lawrencium,for example,with itshalf-life ofeightseconds- remain exotic. There isreallyno "use"for lawrencium.I expect it will likewisebe a whilebeforeMonstergetsmuch ofa workout. But one neverknows, for groups underlie many things: particlephysics, chemistry, and, as Martin Gardner haspointedout, the theorybehindany magictrick involv-ing ropesand twisted handkerchiefs.

It would be a mistake to concludethat mathema-ticians necessarilycare whether a new discoveryhassomepractical application.Mathematicians dowhatthey do because it isbeautiful, interesting,challeng-ing.Whatflaresthe nostrils isthe prospectofa chase.A problem beckons: "Come, Watson. The game isafoot." Still, there are always surprises. Attempts bypractical men to separate the pure from the appliedare artificial concessions to the finitenessof humanthoughtand time.

Number theory, for example, has always beenconsideredthe purest of pure math. It is the study ofwholenumbers,prime numbersin particular. Yet oneof the most practical applicationsof mathematics inrecent years came out of numbertheory: public keycryptography.Thefirst method ofpublic key cryptog-raphy was discoveredby Whitfield Diffieand MartinHellman ofStanford in 1976,but the system I amabout to describe is the work of RonaldRivest,AdiShamir, and LeonardAdleman,all ofMIT.

To understandpublic keycryptography,youmustunderstand the wordmodulo,a conceptintroducedbyKarl Friedrich Gauss in 1801 in his classic Dis-quisitionesarithmetical Theintegers modulofive (togivean example)are obtainedby setting5 equal to 0,so that you count 1,2,3,4,0,1,2,3,4,0,and so on.Any-

thing that is cycliccan be measuredin modulonum-bers. We tell time modulo12. Numerous cardtrickscan be worked modulo 13. Becauseboth theEarth andthe moon follow acyclic pattern,one can usemodulo

Kail Friedrich Gauss( 1777-1855)

from Call lYa-dnch (>.iusv

1777 1^77 />\ Ku,IH,i,h< IV77 b\ firm;

Vf,-os W'rltii;. \1i4>,n h

The WilsonQuarterly Spring 1982103







MATHEMATICS

arithmetic to determinethe date on which Easterwillfall in any year (the method belowwill work foranyyearbetween 1900 and2099).

From "Mathematical Games"by Martin Gardner.(F)798/ byScientificAmerican,Inc. Allrights reserved.

3Indeed, the inte-gersmodulofive withtheoperationof addi-tion is anexampleofa cyclic group ofprime order, whichwe encounteredabove. This is oneofthe nicethingsaboutmathematics. Even asthings get"curiouserand curiouser," one

finds old friends inunexpected places.

Let us go back to modulo five.You can add andmultiplymodulo five(or anyother number)providedyou goback to zeroevery time you reach five. Theeasiestway to castout fives is to divideby 5 and takethe remainder.Thus,4 + 3 = 2 (the remainderwhenyoudivide7 by 5)and 4x4=1 (the remainderwhenyoudivide16by 5).Thisarithmeticmayseemstrange,but it obeys manyof the laws ofordinaryarithmetic.3

What does thishave to do withcryptography?Well,you can see howeasy it is, givena number, tofind out what itequals modulo five. But what aboutgoing the other way? If I tell you what my numberequalsmodulofive,youhavenowayofguessingwhatmy originalnumber is. I tell you my number is 3modulo five.What is it? Itmightbe 8. It mightbe 63.It might even be 3.A kind ofone-waytrap door hasbeenset inplace.This is the basis ofpublic key cryp-

tography,a

tamper-free systemwhose

applicationsrange fromensuringthe privacyof electronic mail torunninga networkofagentsbehind the Iron Curtain.

In practice, public key cryptographyworksmodulo somevery large number (one of 200digits,say)that is theproductof twovery largeprime num-bers. To put a messageinto code,you first transformthe messageinto a number(anyschoolboymethod foraccomplishing this will do) and then raise thatnumber to a power that anyone can know, modulosomevery largenumber that is alsoopen to anyone.Thus, anybodycan encipher a message.But, once a

number has been reduced modulo some othernumber, there is no way of getting the originalnumberback.

Now thecyclic nature of modulo numbers comes

TheWilsonQuarterly Spring 1982104







MATHEMATICS

into play. There is another number, a secretnumber.Whenyouraise theencipheredmessageto this secretpower,it comes around to theoriginal message.

Let's do an actualexample.(Sothat we don't needa computer,I will useunrealisticallysmallnumbers.)The first step is to pick any two prime numbers. Wepick 3 and 11.Multiplythese numbers toget 33. Wewillwork modulo33. Subtract one from eachprimetoget 2 and 10.Multiply2 times10 toget 20. We calcu-lateourpublicnumber and oursecret numbermodulo

20 (since the standard abbreviation formodulo ismod,wesay"mod 20").First,list all the numbersthatdivide 20. Your list should read:1, 2, 4, 5, 10,20.Foryour publicnumber, pick any number between 1 and20 that is not divisibleby any number on this list(except 1, of course). We pick 7. For your secretnumber, use your computer to find a number that,when multipliedby 7 and reduced mod20, gives 1.The mathematics of modulo arithmeticguaranteesthat there will alwaysbe such a number, and in thiscase it is not hard to find. The number 3will do thejob, because 7x3 = 21 and21 mod 20equals 1.So,our secret number is3, and we areready to go.Send your operatives in the field the publicnumber, 7,and tell them all to work mod 33.Keepthenumber 3 sosecret that not evenyouknow it. Thewayto do this is to haveyour computercalculate it, withinstructions to tellno one;but instruct it to raise anynumber entered to this secretpowerand then reducemod 33.

Now,your operativeis ready to sendyou a mes-sage.Because we areworkingwith such smallnum-bers, it must be a very short message.Suppose hedecidesto send the letterB.The easiestwayto changeletters to numbersis to letA = \,B = 2,C = 3,and soon. So, he changes B to 2, raises i% o the seventhpower (7 is the public number, remember) and re-duces mod 33. Two to the seventhpower is 128. Toreduce 128 mod33you divide 128by 33 andtake theremainder,which is 29.Youroperativethen sendsyouthe number29.

Back at the homeoffice,you getthe number. Youfeed 29 intoyour computer,which raises it to these-cret power, 3, and reduces mod33. Thatis, multiply29 times 29 times29, then divideby 33 and takethe

remainder. Doit

andsee

what you get.4With smallnumbers such asthese,it isreally veryeasyto find the secret number.Just factor 33 into 3x11 then find the secret number the same way we

A"cipherdisc"inventedbyGiovanniBattista Porta(1535-1615).

Duvui KuhnColl,;,ion

4Youshouldget

2,whichyieldsthe orig-inal message:B.








MATHEMATICS

BernhardRiemann(1826-66)

Cuion-^atArchivfutKumt unJ(,cschuhh\ Berlin.

found it to begin with. Thekey to the successof thiscipher lies in the fact that it is hard to factor largenumbers. Mathematicians have beentrying to findways to do so for more than 100years. It would takethe fastestcomputernow in existence about76 yearsto factor a200-digitnumber.

Most mathematicians believethat the problemoffactoring very largenumberswill not be solvedin thenear future, but the U.S.National Security Agency(NSA)is takingno chances.The NSA has asked Ameri-

can mathematiciansworkingin the area ofcryptog-raphy to submit all new results togovernmentexpertsin Washington prior to publication. Some mathe-maticiansfind this reasonable. Othersfeel it is anun-warranted intrusion into their lives and work. Forarguments pro and con, see Noticesof the AmericanMathematicalSociety(Oct. 1981).The debatemay beacademic. The NSA ispowerlessto prevent mathe-maticiansin other countries fromconductingresearchin cryptographyand publishingtheir results in, say,the SaskatchewanJournalofNumberTheory.

Let's turn now toanalysis.Analysisis the study of functions.A function isanyrule that assignsa fixedoutput to any giveninput.

Thesquaringfunction is a familiarexample.Input 3,output 9. Input 5, output 25.Other well-knownfunc-tions are thesine functionand the exponentialfunc-tion. Calculus,discoveredindependentlyin the 17thcenturyby both Isaac Newtonand GottfriedWilhelmLeibniz,is the greatestachievement ofanalysis.

To understand the most significantrecent resultin analysis,we need to look at the Riemann zetafunc-tion,named for theGerman mathematicianBernhardRiemann.

TheRiemann zeta function hasapplicationsin algebraand algebraic geometryas well asanalysis.It is used toestimate the number ofprimes in a givenrange (say,between 10 and 10million),which is im-portant in the theoryofpublic key cryptography.

We begin with the number i, the elusivesquareroot of- 1 It wascalledimaginary(in "real life,"youcan't squareanythingandgeta negativenumber),andits existence wasdenied until it proved too useful toignore. Leibniz once called imaginary numbers "awonderfulflight of God'sSpirit; they are almost anamphibianbetweenbeingand notbeing."Ifweutilize

these amphibiansas in the graph on the facingpage,then any number in the field say, 3 + 2i,or -2 - 2i,or even - 2 + Of is known asa complexnumber.Intrigonometry, we learn how todo arithmetic with

TheWilsonQuarterly Spring 1982

106







MATHEMATICS

complexnumbers.The input of the Riemannzeta functionis a com-

plex number. So is theoutput. How is theRiemannfunction defined? To definea function, we need toknowwhat to do with theinput. The squaring func-tion turns 3 (forexample)into 9. Thereciprocalfunc-tion turns 3 intoVa, into %.If thesesimplefunctionsare like light bulbs (input: electricity;output: light),then the Riemannzeta functionis a wholefactory(in-put: raw materials;output: finishedproduct).

The Riemann zeta functioninputs a complexnumber, z,then takesthe series 1+2 + 3 + 4 + 5+.. .("..." means: go on forever), takes its reciprocalterm-by-termto get 1 + %+ % + ..., and then raisesthe wholething, againterm-by-term,to the z power.The sum of thisinfinite series is theoutput of theRiemann

zeta function, and for any z which beginswith anumber greater than 1(e.g.,3 + 2i), the seriesconverges.5This means that it addsup to a finitenumber. (The series 1 + Vi + V4+ % + Vi6 . . alsoconverges:It adds up to 2. While somepeople maywonder how one canactually add up an infinitenumber of terms- Will weeverreallyreach2? IsaacNewtondiscovereda method:calculus.)

Now, whenever you have a function, a naturalquestionto askis, "Whereis the output zero?" Thisiswhat most ofhigh schoolalgebra was about. Whereare the zeroes of x2+ 2x- 3? Answer:x = -3, or x = 1

Where are the zeroes of the Riemann zeta function?Todate, 3.5 million zeroes have been foundby computer,not counting the zeroes (considered trivial)that liealongthe horizontal axis. Sofar, all of the 3.5million

5For z's that beginwith a number lessthan orequalto 1 wemust extendthe zetafunctionby a processknown as analyticcontinuation. Oncethis is done, then forevery complex num-ber except1 + Oi,theRiemann zeta func-tionconverges.








MATHEMATICS

ian space-timeis an exampleof a spacethat has fourdimensions.)

A one-dimensionalmanifold is a curve. A two-dimensionalmanifold is a surface. Wemight call athree-dimensional manifold a three-space, and afour-dimensionalmanifold ahyper-space.It is ofteneasier topicture a manifoldby putting it in a Eucli-dean space of higher dimension.A sphere,for exam-ple, is simply a surface:a two-manifold.But try topicture a sphere without picturing it in three-

dimensional space. Mathematically, that three-dimensionalspaceis unnecessary,but conceptuallyitis essential.7

The question then arises: What is the smallestdimensionalEuclideanspacein whichyoucan put agiven manifold?In 1944,Hassler Whitneycame upwith a preliminary answer. He proved that any«-dimensionalmanifold(n-manifold,forshort)can beput at least into 2n-l dimensionalEuclideanspace.So, we can always put a three-manifold into five-space. One-and two-manifoldsare well understood,but there is a profusionof three-manifolds,and theyare extremelyhard to get a grip on. Seeinghow theyfit into Euclideanspaceis abig help.

Whenwe put one space into another, we do notwant totear it or creaseit. Forexample,we canflattena sphere,but not withouta sharpcreaseat theedge,sowe cannotput a sphere into a plane. There are twolegitimate ways of putting one space into another:immersionand embedding.An immersionallows thespaceto intersectitself,an embeddingdoesnot.Afig-ure eight is an immersionof a circle in the plane: Itintersects itself. The letter O is an embeddingof a

circlein the

plane.To the

rightis a

pictureof aKlein

bottle; it is an immersionbecausewe constructit bypassingthe neckof thebottle throughthe bottle'sside.

The new theorem, proved by Ralph Cohen ofStanford,and conveyedto meby Don DavisofLehigh,shows thatany n-manifoldcan be immersedin Eucli-dean spacewhose dimensionis In minus thenumberof l's in abinary expansionof n.Thebinary systemisa way of expressingany number in terms of l's andO's,and the transformation of an ordinary numberinto binary form is easily accomplished(seebox onnextpage).Thus,any five-manifoldcan be immersed

in Euclideaneight-spacebecausefivein binary formhas two onesin it.A topologicalquestion that remains unsolvedis

the dimension of the smallest Euclidean space in

7There are three-dimensionalspheres,four-dimensionalspheres, and so on,but they are difficultto picture.If youhadtwo balls and wereable to glue theirskinstogetherso thatone sphere wasturned insideout overthe other, then you

would have a three-dimensionalsphere.

From Graduate Textsin Mathematics byJohn Stillwell.

(c)1980by Springer-Verlag.NewYork.

TheWilsonQuarterlySpring1982109







MATHEMATICS

n between 3 and30,000.But such computersearchesgiveno insight into ageneralsolution to theproblem.Wemaynever knowwhetherFermât s last theoremiscorrect,but the effortsto proveor disproveit overthecenturies have led to much usefuland ingeniousmathematics- a reward in itself.

Themost important unsolvedproblemin mathe-matics may be the Poincaréconjecture, posed bythebrilliant French astronomer-mathematician Jules-Henri Poincaré.To understand it, one must return tothe topologicalspacescalled manifolds.8A manifold istermed "simply connected"if anyloop of thread on its surface can be pulled in - bysomeoneholdingboth ends of theloop firmly- whileat the same timeremaining in continuous contactwith the manifold. Asphereis simplyconnected;try itwith a pieceof thread and a rubber ball.Atorus is notsimplyconnected.Aloopof thread around itwill losecontact with the surface as itis pulled in and passesover thehole; if the thread goes throughthe hole tobegin with, one cannotpull it in at all. The Poincaréconjecture states that the only simply connectedthree-manifold is thethree-dimensionalsphere.A version of the Poincaréconjecture has beenproved for dimensions five andabove (i.e., the onlysimply connected five-manifold is the five-dimen-sional sphere). The four-dimensionalcase has justbeenproved byMichaelFreedmanofthe UniversityofCalifornia,San Diego.(Anothermportant newresult.)Thethree-dimensional caseremains unsolved.

But there are distant rumblings.Mathematiciantalks to mathematician. Whilenothing has yet ap-peared in print, the word is, to everyone'ssurprise,that the Poincaré

conjecture maybe false.

So it goes, new mathematics from old, curvingback, foldingand unfolding,old ideasin new guises,new theorems illuminating old problems. Doingmathematics is likewandering througha new coun-tryside. We see a beautifulvalley below us, but theway down is too steep, and so we take anotherpath,which leads usfar afield, until,by a suddenand unex-pected turning, we find ourselveswalkingin the val-ley, admiringthe treesand flowers.

Jules-HenriPoincaré(1854-1912)

From Major Prophet»of TodaybyEdwinE. Sbssen.©1914

byLittle,Brown,ndCompany.

«What follows con-cerns manifoldsthatare "finite" in a spe-cial technical sense ofthe word that we donot needto go into.Amathemetician wouldcall them "closed,connected" mani-folds.

The WilsonQuarterlyISpring 1982111

art - mathematics, monsters & manifolds

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