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The Science of Patterns

LYNNARTHUR STEEN

forcesoutside while contributingto humancivilization The rapid growth of computing and applicationshas a rich and ever-changingvariety of intellectualflora and fauna. helpedcross-fertilize the mathematicalsciences, yielding These differencesin perceptionare due primarilyto the steep and an unprecedentedabundance of new methods,theories, harshterrain of abstractlanguage that separatesthe mathematical andmodels. Examples from statisiicalscienceX core math- rainforest from the domainof ordinaryhuman activity. ematics,and illustrate these changes, The densejungle of mathematicshas beennourished for millennia which have both broadenedand enrichedthe relation by challengesof practicalapplications. In recentyears, computers between mathematicsand science. No longer just the have amplifiedthe impact of applications;together, computation study of and space, mathematicalscience has and applicationshave swept like a cyclone across the terrainof become the science of patterlls, with theory built on mathematics.Forces -unleashed by the interactionof theseintellectu- relations among patterns and on applicationsderived al stormshave changed forever and for the better the morpholo- from the fit betweenpattern and observation. gy of mathematics.In their wake have emergednew openingsthat link diverseparts of the mathematicalforest, making possible cross- fertilizationof isolatedparts that has immeasurablystrengthened the whole. MODERN MATHEMATICSJUST MARKED ITS 300TH BIRTH- Throughoutthe 20th century,mathematics has grown rapidlyon day. The publicationin 1687 of Newton'sPrincipia manyfronts. The classicalcore has remainedrooted in the Newtoni- Mathematicaestablished mathematics asthe methodolog- an mathematicsof ,a synthesis of algebraand icalparadigm of theoreticalscience. Newton perceived patterns in appliedto the studyof how things change.But even as this core has the accumulatedastronomical of his time;he abstractedfrom expandedunder explosivepost-World War II growth, it has been thesepatterns certain general principles (whence Principia); then he supplementedby major developmentsin other mathematicalsci- usedthese principles to deducepatterns both known and unknown ences in , logic, , , in the behaviorof planetarybodies. His wasa scienceof patterns- ,computation, , and ,in rootedin data,supported by deduction,confirmed by observation. to ,geometry, and analysis. By the endof the 19thcentury, Newton's creation had flowered In each of these subdisciplines,applications parallel theory. Even magnificently,producing unprecedented intellectual blossoms. Eu- the most esotericand abstractparts of mathematics numbertheory ropeangiants such as Euler, Lagrange, and Weierstrass had elaborat- and logic, for example arenow used routinelyin applications(for ed andrefined the , establishing the foundations for modern example,in computerscience and ).Fifty years ago G. analysis.James Clerk Maxwell used Newton's to write H. Hardycould boast of numbertheory as the most pure and least the lawsof ,and GeorgBernhard Riemann ap- usefulpart of mathematics(1); todaynumber theory is studiedas an plieddifferentials to geometryin apt(albeit unintentional) prepara- essentialprerequisite to manyapplications of coding, includingdata tion forAlbert Einstein, who soonwould discover in Riemanniantransmission from remote satellites,protection of financialrecords, geometrythe keyto a generaltheory of gravitation. and eflicientalgorithms for computation. At the sametime, on a separatecontinent, the peopleof the In 1960, at a time when theoreticalphysics was still the central UnitedStates were beginning their second century without mathe- jewel in the crown of appliedmathematics, Eugene Wignerwrote maticalor scientificgiants in theirmidst. Yet in 1888,two centuries aboutthe "unreasonableeffectiveness" of mathematicsin the natural afterPrincipia,a fewfar-sighted individuals foundedwhat is nowthe sciences:"The miracleof the appropriatenessof the languageof AmericanMathematical Society, thereby setting in motiona process mathematicsfor the formulationof the laws of physicsis a wonder- that createdthe world'sstrongest environment for mathematicsful gift whichwe neitherunderstand nor deserve"(2, p. 14). Indeed, research.In recognitionof thisanniversary, American mathematics theoretical has continuedto adopt (andoccasionally invent) is now celebratingits 100thbirthday. increasinglyabstract mathematical models as the logicalfoundation for currenttheories: Lie groups and gauge theories,exotic expres- sions of symmetry,have joined fermionsand baryonsas fundamen- tal tools in the physicist'ssearch for a unified theory of both Forcesfor Change microscopicand macroscopicforces of nature. Manyeducated persons, especially scientists and engineers, harbor Duringthis sameperiod, however, striking applications of mathe- an imageof mathematicsas akinto a treeof knowledge:formulas, matics emergedacross the entire landscapeof natural,behavioral, theorems,and results hang like ripe fruits to be pluckedby passing and socialscience. Moreover, applications of one partof mathemat- scientiststo nourishtheir theories. Mathematicians, in contrast, see ics to another-of geometryto analysis,of probabilityto number theirfield as a rapidlygrowing rain forest, nourished and shaped by theory provide renewed evidence of the fundamentalunity of mathematics.Despite the ubiquityof connectionsamong problems The authoris professorof mathematicsat St. Olaf Collegein Northfield,MN 55057 in science and mathematics,the discoveryof new links retains a and chairmanof the ConferenceBoard of the MathematicalSciences. surprisingdegree of unpredictabilityand serendipity.Whether

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This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions plannedor unplanned,the cross-fertilizationbetween science and ideasrooted in antiquity.Its tools areabstraction and deduction; its mathematicsin problems, theories, concepts, and paradigmshas edificesinclude functions, equations, operators, and infinite-dimen- never been greaterthan it is now, in the last quarterof the 20th sional spaces.Within core mathematicsare found the traditional century. In 1988 one can say with some justificationthat the subjectsof numbertheory, algebra, geometry, analysis, and topolo- effectivenessof mathematicsis even more "unreasonable"than ever gy. After a half-centuryof explosive specialized growth, core before. mathematicsis experiencinga renaissanceof renewedintegrity based Parallelingthe growingpower of applicationsof mathematicshas on the unexpectedbut welcome discoveryof deep links among its been the extraordinaryimpact of computing. It is ironic but variouscomponents. inuisputablethat computerswere made possible by applicationof Appliedmathematics fits mathematicalmethods to the observa- abstracttheories of mathematicianssuch as Boole, Cantor,Turing, tions and theoriesof science.It is a principalconduit for scientific andvon Neumann,theories that just a few decadesago werewidely ideas to stimulatemathematical innovation and for mathematical deridedby criticsof the "new math"as wild abstractionsirrelevant tools to solve scientificproblems. Traditional methods of applied to practicalpurposes. It is doubly ironic that the computeris now mathematicsinclude differential equations, numerical computation, the most powerfulforce changingthe natureof mathematics.Even controltheory, and dynamical systems; such traditional methods are mathematicianswho never use computersmay frequentlydevote today being appliedin majornew areasof applications,including their entireresearch careers to problemsgenerated by the presence combustion,turbulence, optimization, physiology, and epidemiol- of computers.Across all parts of mathematics,computers have ogy. In addition,new tools fromgame theory, decision science, and posed new problemsfor research,provided new tools to solve old discretemathematics are being appliedto the humansciences where problems,and introducednew researchstrategies. choices,decisions, and coalitionsrather than continuous change are Althoughthe publicoften views computersas a replacementfor the apt metaphorsfor descriptionand prediction. mathematics,each is in realitya power tool for the other. Indeed, All attempts to divide mathematicsinto parts are necessarily just as computersprovide new opportunitiesfor mathematics,so artificialand perpetuallyin flux: statisticalscience, core mathemat- also mathematicsmakes computers so incrediblyeffiective. Mathe- ics, and appliedmathematics represent just one of many possible maticsprovides abstract models for naturalphenomena, as well as structuresthat may help one understandthe whole. Thesedivisions algorithmsfor implementingthese models in computerlanguages. do not representintrinsic differences in the natureof the discipline Applications,computers, and mathematicsform a tightly coupled so muchas differencesin style,purpose, and history; they may more systemyielding results never before possible and ideasnever before aptlydescribe types of mathematiciansthan types of mathematics. . * . lmagmea. Othershave attempted to portraythe natureof modernmathematics in somewhat differentterms [see, for example, (3-5)]. What is importantabout these labels is that they help focus attentionon certaincharacteristics of the mathematicalsciences, not that they The MathematcalSciences themselvesrepresent inherent or essentialcompartments. Rapid growth in the nature and applicationsof mathematics One featurethat is inherentin mathematicsand is essentialto meansthat the Newtoniancore calculus,analysis, and diffierential understandits role in scienceis this: today'smathematical sciences equations is now just one part of a more diverse mathematical arevery differentfrom what they were a quarter-centuryago when landscape.Yet most scientists have explored only this original most of today's scientists last studied mathematics.Computers, territory,because that is all that was includedin theircurriculum in applications,and cross-fertilizationhave combined to transformthe high school, college, and graduateschool. With the exceptionof mathematicalsciences into an extraordinarilydiverse and powerful statistics,an old sciencewidely used acrossall disciplinesthat has collection of tools for science. Without even asking permission, become largelymathematical during the 20th century,the narrow mathematicianshave quite literallyrebuilt the foundationsof sci- Newtonian legacy of analysisis the principalconnection between ence. The work is not finished, but its new shape is sufficiently practicingscientists and the broadmathematical foundations of their visiblethat all who use mathematicsshould take the time to explore disciplines.The dramaticchanges in the mathematicalsciences of the its new features. last quartercentury are largelyinvisible to those outside the small communityof researchmathematicians. Today's mathematicalsciences, like yesterday'sGaul, can be StatisticalSciences divided into three parts of roughly comparablesize: statistical science,core mathematics,and appliedmathematics. Each of these Both computersand statisticsdeal with data: what computers three majorareas is led (in the United States) by a few thousand record,transform, and manipulate,statisticians interpret, summa- activeresearchers and receivesapproximately $50 millionin federal rize, and display.This confluenceof problemsource with problem researchsupport annually. Although the boundariesbetween these solverhas radicallytransformed (some would sayrestored) statistical partsoverlap considerably, each provincehas an identifiablecharac- science to a data-intensivediscipline. Phenomena described by ter that correspondswell with the three stagesof the mathematical traditionalstatistical distributions (normal, Poisson, and so forth) paradigmestablished by Newton: data,deduction, and observation. representonly a tiny partof the enormousquantity of datacaptured Statisticalscience investigates problems associated with uncertain- by computersall over the world. Here are three examplesof what ty in the collection,analysis, and interpretationof data.Its tools are the dominanceof data has done in the statisticalsciences. probabilityand inference;its territoryincludes stochastic modeling, Spatial statistics.The increasinguse of electronicscanning devices statisticalinference, decision theory, and experimental design. Statis- (forexample, in tomography,airborne reconnaissance, and environ- tical science influencespolicy in agriculture,politics, economics, mentalmonitoring) has producedan urgentneed for sophisticated medicine,law, science, and engineering.Advances in instrumenta- analysisof datawith inherentspatial structure. Image enhancement, tion and communication(to gatherand transmitdata) have posed the visual clarificationof blurredimages, is the most common new challengesto statistics,leading to rapidgrowth in new methods application.Other important tasks include the visual representation and new applications. of datato enablean observerto detecthidden patterns and the Core mathematicsinvestigates properties of numberand space, statisticalcompression of datain realtime to permitefficient storage

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This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions and subsequentanalysis without loss of importantinformation. plenarylectures surveying the currentstate of mathematics,more Researchin spatialstatistics utilizes a wide varietyof mathemati- thanhalfwere on topicslinked in some waywith computation.Here cal, statistical,and computationaltechniques. Problems of separat- aresome examples. ing signalfrom noise borrowtechniques from engineering; ill-posed Bieberbachconjecture. Louis de Branges of Purdue University scattering problems use methods of numerical ; discussedhis proof of the 70-year-oldBieberbach conjecture con- smoothingof data requiresstatistical techniques of regularization. cerning the size of coefficientsin the power series expansionsof And underlyingall this is the inherentgeometry of the problem, certainanalytic functions of a complexvariable (9). At a crucialstage which in many casesis dynamicand nonlinear. in the proof, de Branges had reduced the entire argumentto Nonparametricmodeling. The most fillly developedpart of tradi- verifyingan inequalitybetween two :this was done by tional statistics involves models based on assumed distributions computerto a sufficientdegree to provideconvincing circumstantial determinedby a smallnumber of continuouslyvarying parameters. evidenceof the validityof this of argument.The finallink in the Yet most real data (for example, census informationor satellite formal argument was supplied by a theoretical proof of this imaging) are a mixtureof variablesthat are partlyparametric and ,actually known and proved in the theory of special partly nonparametricor partly continuous and partly discrete. functionslong beforede Brangesneeded it. Withoutplausible a prioriassumptions concerning the distribution Factorinf inte,gers.Henrick W. Lenstra of the University of of nonparametricdata, traditionalstatistical methods were often Amsterdamapplied to one of the oldest prob- unreliable although still frequentlyused. Now, however, large lems in mathematics how to find the prime factorsof an data sets of mixed parametricand nonparametricvariables make it (10). Integerfactorization moved from a backwaterto high priority possiblefor statisticiansto use computationallyintensive methods to in mathematicssimply because of its applicationin computer-based estimatestatistics (for example,regression coeEcients) with reliable cryptography:a code based on the product of two large prime errorbounds for nonparametricvariables. (Of course,the computer numberscannot be brokenwith currentalgorithms because there is has madesuch methods not only possiblebut necessaryby enabling no known eEcient method of recoveringthe two factors from scienceto inundateus with massivedata sets.) knowledgeonly of the product. Bootstrapand yackknifestatistics. Many applicationsof statistics Lenstraattacked the problemof factoringby using ellipticcurves, (notablysurvey analysis and clinicaldata from innovativemedical the of zeroes in a suitableprojective plane of cubic protocols)involve small data sets fromwhich one would liketo infer equationsin threevariables. Points on these curvesform an Abelian meaningful("significant") patterns. Bradley Efron at StanfordUni- group, whose propertieslead directlyto the fastest algorithmyet versityhas pioneeredan innovativemethod of using limiteddata to discoveredfor factoringlarge .Similar analyses are being generatemore datawith the samestatistical characteristics (whence carriedout for factoringpolynomials efficiently (11). Ellipticcurves, "bootstrap"methods). the primarytool in these new ,are the centralobjects In particular,bootstrap methods use computationalmethods to in.volvedin Gerd Faltings's1983 proof of the Mordell conjecture resamplethe given data repeatedlyin orderto generatemillions of (12), surely one of this decade's most stunning mathematical similarpossible data sets, which yield accurateapproximations to achievements. variouscomplex statistics. By comparingthe valueof these statistics Computationalcomplexity. Arnold Schonhageof the Universityof for the given sample with the distributionobtained by repeated Tubingen applied the theory of computationalcomplexity the resampling,one can determinewhether the observed values are analysisof the inherentdifficulty in solving problems to the basic significant(6). Jackknifemethods are related to bootstraptech- taskof solving equations,beginning with the simplest,c = b, for niques,but the way they reducebias in the statisticalprocedures is to ordinarynumbers (13). The traditionalsolution, , has not repeatedlyslice awaypart of the data. beenof interestto mathematicianssince the MiddleAges, yet now it is a subjectof intenseresearch. Only recentlyhas it been provedthat the known methodsof performingdivision of complexnumbers is Core the best way possible in the sense that no method can involve Mathematics fewerreal arithmetical operations. The methodsused in this funda- The forcesthat impingeon mathematics primarilyapplications, mentalanalysis of arithmeticpresage similar analysis for the entire computers,and cross-fertilization influencethe core ("pure")parts rangeof computeralgorithrns, enabling mathematicians to deter- of the subject in profound ways. To illustratethe nature of the mine limits of computationalfeasibility as well as areas ripe for changes,I will use two ratherdifferent areas of impact,computation furtherimprovement. andgeometry, as widelyseparated mileposts along a vast continuum Iteratedmaps. Stephen Smale of the Universityof Californiaat of mathematics. Berkeleyexamined the classicalproblem of findingzeroes of polyno- Core mathematicshas changedunder the influenceof computers mialsin the verymodern context of iteratedmaps and computation- as much as the more appliedareas of the mathematicalsciences, but al complexitytheory. The prototypicalexample is Newton'smethod in differentways. Most noticeableis the shift in researchinterests to for Snding a zero of a function/(x): questions motivated by computation. But computers have also changedthe way conjecturesare invented and tested, the way proofs Xn + 1 -Xn -f(xn)lf (Xn) are discovered,and in an increasingnumber of cases the nature In the pre-computerstyle, this method was appliedone at a of proof itself. time, beginningwith a reasonablygood guess so that the sequence The archetypeevent in computer-assistedmathematics was the of points convergedrapidly to a zero of the function. 1976 proof of the century-oldfour-color conjecture,which was Smalestudied Newton's method globally, looking at the mapping basedon a computeranalysis of thousandsof reductionpatterns to of the entire complex plane onto itself generated by Newton's bridge a gap between mathematicaltheory and human analysisof methodfor a particularpolynomial (14). The distributionof points cases.This eventshook the veryepistemology of mathematics(7, 8) . that eventuallymap to zero is an exampleof a "fractal"set that Ten yearslater, in 1986, the InternationalCongress of Mathemati- displays many of the chaotic properties typical of turbulence. cians set forth the state of worldwide mathematicsat a 10-day Smale'sanalysis applied global iterationto the simplexmethod of conferenceat the Universityof Californiaat Berkeley.Of the 16 solving linear programmingproblems, leading to a proof of the

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This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions empiricalfact that, on average,the simplexmethod behavesvery problems:new minimalsurfaces have been found with the aid of well indeed.His methodsalso leaddirectly to new modelsfor chaos computergraphics, and the visualdisplays of iterativemaps (in the that have extensiveapplications in physicsand (15, 16). widely known "fractal"pictures) make visible patternsthat would never have been noticed by analyticmeans alone. Fractalpatterns have provided an apt description of a large class of physical phenomena,ranging from fracturingof glass to texturein surfaces Geometry (18-20). If computerstypify the modern era of mathematics,geometry Geometriccomputing is beginningto prove very useful also in epitomizesits classicalroots. Historically,geometry, the study of core areasof mathematicsremote from geometry,because super- space,has been one of the majorpillars of core mathematics.For computerscan calculate and display in visualform roots of equations variousreasons, its role in the mathematicscurriculum has declined andother mathematical objects. This enablesmathematicians for the over the past 20 years,so that even those with universitydegrees in firsttime to "see"the content of the abstracttheorems they prove mathematicsoften havelittle acquaintancewith geometrybeyond an and therebyto make new conjecturessuggested by the eye rather archaicand typicallyrigid encounterwith Euclideanproofs in high thanby the mind. (In recognitionof this new frontier,several of the school geometry.In sharpcontrast to this curriculardecline is the world'sleading geometers have recentlyreceived fimding to estab- renaissanceof geometry in researchmathematics. In a very real lish a GeometrySupercomputer Project to carryout researchin this sense, geometryis once againplaying a centralrole on the stage of areain both the United Statesand Europe.) mathematics,much as it did in the Greekperiod. A principalactor in the study in modern geometryis a "mani- fold,"a termused by geometersto describesurfaces and spacesthat arelocally like . form the naturallocus for AppliedMathemadcs solutions to differentialequations, and in turn their geometry Appliedmathematics is distinguishedfrom core mathematicsnot imposes structureon the analyticnature of these solutions. Thus so muchby contentor methodas by objective:in appliedmathemat- manifoldsare of importancenot just to geometrybut to all partsof ics, valueis measuredby the degreeto whichnew methodsimprove classicalanalysis. scientificunderstanding or technologicalapplications. Two of the three 1986 Fields Medals the "Nobel prizes"of The roots of the scientificrevolution lie in the introductionby mathematics went to Michael Freedmanof the University of Galileoof empiricalmethods to replacethe speculativeexplanations Californiaat SanDiego arsdSimon Donaldson of OxfordUniversity of classicalGreek natural philosophy. Newton introducedtheoreti- for work in the geometry of four-dimensionalmanifolds (17). cal science by showing that empiricaldata can be explainedby Freedmanshowed that the Poincareconjecture concerning spheres mathematicalresults deduced from basicaxioms. In our time, John is true for four-dimensionalmanifolds, thereby resolving the next- von Neumannpioneered the computationalparadigm in which the to-last case of this 80-year-oldconjecture. (Only dimensionthree resultsof theoreticalscience are used to simulatereality on a modern remainsunsettled.) Freedman's methods showed that the topologi- computer.As a consequence,computational methods now pervade cal classificationoffour-dimensional manifolds mimics the algebraic all aspectsof appliedmathematics (21, 22). classificationof quadraticforms. Necessityis the mother of invention,in mathematicsas in life. Donaldsonused the Yang-Millsfield equationsof mathematical Becausethe needsof sciencestimulate the growthof mathematicstas physics,themselves generalizations of Maxwell'sequations, to study scienceexpands and grows so does mathematics.The consequence instantonsin four-dimensionalspace, thereby reversing tradition by has been an explosivegrowth in the natureand range of applied applying methods from physics to the understandingof .Four very differentareas will serveto illustrateboth mathematics.By exploitingthose propertiesof differentialequations the diversityand the innovationof presentresearch. that reflectthe wave-particleduality of matter,Donaldson was able Biolo,gicaZsciences. Nothing better illustratesthe potential for to develop an entirelynew approachto the study of fundamental mathematicsin the biological sciences than the many traces of problemsof geometry. mathematicsbehind the Nobel prizes (23). For example,the 1979 One consequenceof Freedman'sand Donaldson'swork was the Nobel Prize in medicine was awardedto Allan Cormackfor his discoverythat in four dimensionsthere aredifferentiable manifolds applicationof the Radon transformta well-knowntechnique from that are topologically but not differentiablyequivalent to the advancedclassical analysis, to the developmentof tomographyand standardEuclidean four-dimensional space. Such "exotic" spaces are computer-assistedtomography (CAT) scanners.One of the recipi- unique to four, which also happensto be the natural ents of the 1985 Nobel Prizein chemistrywas biophysicistHerbert domainof the space-timecontinuum of our physicalworld. Wheth- Hauptman,who took his Ph.D. in mathematicsand who is presi- er these uniqueproperties are accidentalor significantis something dent of the MedicalFoundation of Buffalo.Hauptman was citedfor thatwill requiremuch further investigation. Indeed, the geometrical fundamentalwork in Fourieranalysis pertaining to x-raycrystallog- theorycreated by this work has alreadyled to applicationsin string raphy. theory, therebyfeeding back into physicsspinoff benefitsof ideas Indeed, recent researchin the mathematicalsciences suggests originallyborrowed from physics.: dramaticallyincreased potential for fimdamentaladvances in the life Compter,graphics.Geometry and computersintersect in one of scienceswith methods that depend heavily on mathematicaland the most lively and attractiveinterstices of the mathematicalsci- computermodels. Structuralbiologists have become genetic engi- ences: computer graphics. Although most well-known uses of neers,capturing the geometryof complexmacromolecules in super- computer graphics are in areas of applied mathematics,visual computersand then simulatinginteraction with other moleculesin techniquesare making a realimpact in traditionalcore mathematics theirsearch for biologicallyactive agents. Using these computation- as well as in the teachingof mathematicsat all levels. al methods, biologists can portray on a computer screen the To producerealistic graphics on a computerrequires considerable geometryof a cold virus an intricatepolyhedral shape of uncom- theoreticalinteraction among geometrical representation, algebraic mon beautyand fascinatinggeometric features and searchits encoding,and computer algorithms. In return,computer graphics surfacefor molecular footholds on whichto securetheir biological methodshave providedcrucial assistance in manymathematics assault.

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This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions Xs

Geneticistsare beginning the monumentaleffort to mapthe entire cases,key variablesare totally unknown or areobscured by noise. A human genome, an enterprisethat requiresexpertise in statistics, common case, representedmathematically by what are known as combinatorics,artificial intelligence, and data managementto or- stochasticdifferential equations, occurs when the systemis subjectto ganize billions of bits of informationcoming from all over the externalwhite noise. Typicalexamples include diffusion processes in world. Ecologists,the firstmathematical biologists, continue to use communicationtheory, chemicalprocesses, stock marketanalyses, the extensivetheories of populationdynamics to predictthe behav- epidemiology,queueing theory, and populationgenetics. ior and interactionof species, taking into account such realistic All these examplesshare certain common features.Most impor- complicationsas sex-specificmortality, reproductive , and tant, like a faircoin whose probabilityof headsdoes not dependon predator-preydata. Neurologistsnow use the theory of graphsto the outcome of previous tosses, they are processes without a model networksof nervesin the body and the neuraltangle in the memory of the past. In mathematicaljargon, they are what are brain. And, finally, physiologists use contemporaryalgorithms knownas Markovprocesses. In addition,they aresubject to random appliedto l9th-centuryequations of dynamicsto determine influence of key variables whence the stochastic nature of the such things as the effects of turbulencein the blood caused by problem. swollenheart valves or clumpsof cholesterol. Stochastic differentialequations are one of the key areas of ControZtheory. One of the most widespreaduses of mathematical probabilitytheory. Recent researchhas yielded importantlinks to techniquesis in the control of systems; applicationsrange from other parts of mathematics.In particular,certain statistics of the quality control on an assemblyline to flight control of a high- stochasticsystems (for example, mean exit times of a diffusion performanceaircraft. is one of the manymathemati- process)turn out to be solutionsof ordinary(nonstochastic) partial cal products of World War II, and until recently it has been differentialequations. This in turn has yielded interestinginsights dominatedby the originalparadigm of a single input-singleoutput linkingrandom processes in one systemwith geometryand analysis system that can be representedmathematically as a single-valued in another,notably . function of one .Such models are sufficientlysimple to The behaviorof the solar systemunder the randominfluence of permitan experiencedengineer to optimizesystem performance by passing comets and stars is an example of a dynamicalsystem adjustingparameters in the model by trial and error. governedby a stochasticdifferential equation. Since randomdistur- However,the adventof computer-controlledsystems has opened bancesmay destabilizedynamical systems (even if the disturbances up a new andvery complex frontier of systemswith manyinputs and are small), the investigation of stability is a matter of utmost many outputs. The prototypicalexample of enormouscomplexity importance.Only recentlyhave researchersbeen able to determine for such systemsis the design of very largescale integrated circuits. conditionsunder which the trajectoriesof solutions to stochastic The problemof finding efficientcontrol laws for such systemshas flows will cluster in stable patterns. [Some researchersthink the been approachedby means of the Kalmandigital filter and, more stock marketis subjectto the same types of instability,in which recently,by an extensionof interpolationtheory to matrix-valued behaviornear a "strangeattractor" can leadto sharposcillations that analyticalfunctions and to a full calculusof operators. are inherentlyunpredictable (24).] One benefitof the new theorieshas been to increasethe flexibility Physics.Reconciling quantum theory with general relativ- of control system design. Insteadof being restrictedto traditional ity the physics of the small with the physics of the large is methods introduced by Norbert Weiner that minimized mean perhapsthe majortheoretical problem in physics. Becauseof the square error, designers can choose, with these new systems, to enormous difference in scale between gravitationaleffects and minimizeworst case errors.Such options are of crucialimportance quantumeffects, realistic experiments are of littlevalue in suggesting when preventionof failure(for example,in nuclearpower plants or avenuesfor exploration. in aircraftcontrol) is of primaryimportance. Such benefits will come The leadingcurrent attempt at reconciliationof microcosmicwith as circuit design shifts from scalarto matrix patterns,based on macrocosmicphysics is the theory of strings,a constructin which highly sophisticatedmathematical theories of interpolation. the dimensionlesspoints of four-dimensionalspace-time are re- The availabilityof computersto carryout complexcalculations placed by thin strings in ten , where the intrinsic has opened up many additionalfrontiers in control theory. For structureof the extradimensions, like that of whatwe perceiveto be example,the theoreticalbasis for reconstructingan optimal signal "empty"space, occurs at a scaletoo smallfor our perception.Pursuit from distortedversions receivedby multiple detectorsuses tech- of this hypothesis(which can probablynever be tested by experi- niques from the theory of analyticfunctions in severalcomplex ment) leadsnaturally to higherdimensional symmetries in which are variables.New projective (Karmarkar-like)algorithms for linear embeddedthe special symmetries,the fundamentalinvariants, of programminghave made possible automated control of rapid both macro-and microscopicphysics (25). processes(for example,high-performance aircraft) not previously Knot theory.Fifty years ago mathematiciansdeveloped the theory possible.Certain nonlinear control problems, such as the motion of of operators,motivated in part by the need to find mathematical an airplanewhile engagedin shortlanding and takeoff,can often be models of quantum .Operator theory flourishedas a transformedvia Lie algebrasinto a linearfeedback law that can then *branch of functionalanalysis, which was pursuedboth for its pure be computed with the use of traditionalmethods that are both mathematicalinterest and for its continuedapplicability to quantum computationallystable and sufficientlysimple to carryout in real physics. In recent years, investigationof new types of operators time. yieldednew methods of classificationthat in turn have led to the Stocholsticdiferential eqxations. The laws of natureare expressed discoveryof importantrelations between operator and the most eloquentlyin the languageof differentialequations. Maxwell's classificationof knots, a vexinglydifficult problem that had previ- equationsfor electromagneticfields, Newton's equationsfor plane- ously defiedall attemptsat solution. tary motion, and Navier-Stokesequations for fluid flow are as The key to classificationof knots is a to encode knot articles in nature's constitution. They express the way nature patternsin algebraicterms so that algebraicmanipulations corre- changesin termsthat makepossible both mathematicalanalysis and spond to physical actions on the knot (26) 27). This makes it sacntlnc mvestlgatlon. possible,for the firsttime, to determinewhether one knotcan be But in practice,the data availableto a scientistare never known transformedinto another or unlinkedcompletely into a straightline. exactlyand may be subjectto significantrandom variations. In some Underlyingboth the newtypes of operatorsand the newclassifica-

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This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions tionof knotsis a newalgebraic structure whose characteristics were amongsubdisciplines of mathematics. revealedby its appearancein knot groups,in certainareas of Becauseof computers,we seemore than ever before that mathe- statisticalmechanics, and in exactlysolvable models. As happensso maticaldiscovery is likescientific discovery. It beginswith the search oftenin mathematics,the significanceof thenew structure emerged forpattern in data-perhapsin numbers,but often in geometricor in recognitionof its ubiquity:that the samepattern appeared in algebraicstructures. Generalization leads to abstraction,to patterns severalplaces is preciselywhy the techniquetakes on special in the mind.Theories emerge as patternsof patterns,and signifi- power. canceis measuredby the degree to whichpatterns in onearea link to Recentlybiologists studying the replication of DNA haveteamed patternsin otherareas. Subtle patterns with the greatest explanatory up with mathematiciansworking in knottheory because DNA in powerbecome the deepest results, forming the foundation for entire thecell is normallycoiled into a tightknot. How DNA canreplicate subdisciplines. andthen pull apart if it is tightlyknotted is difficultto imagine like Texasphysicist Steven Weinberg, echoing Harvard mathemati- the magician'strick of effortlesslyseparating two intertwinedrope cianAndrew Gleason, suggests that the reason why mathematics has knots.From motivation and applicationin quantummechanics the uncannyability to providejust the rightpatterns for scientific throughesoteric research in pure mathematicsand then to the investigationmay be becausethe patterns investigated by mathema- unfoldingof DNA is anamazing, albeit not atypical,example of the ticiansare aU the patternsthere are (23). If patternsare what manyinterconnections among diverse parts of mathematics. mathematicsis all about,then the "unreasonableeffectiveness" of mathematicsmay not be so unreasonableafter all.

Patterns REFERENCES AND NOTES These examplesfrom contemporarymathematical science illus- 1. G. H. Hardy,A MathematiciannsApology (Cambridge Univ. Press, Cambridge, trate metaphorsof the mathematicalmethod that originated300 1940). 2. E. P. Wigner,Commun. Pure Appl. Math. 13, 1 (1960). yearsago in the Newtoniansynthesis: data, deduction, and observa- 3. RenewingU.S. Mathematics: Cntical Resource for theFuture (National Academy of tion. They also revealthe effectsof the majorforces for changein the Sciences,Washington, DC, 1984). mathematicalsciences: computers, applications, and cross-fertiliza- 4. MathematicalSciences: A Unifyingand DynamicResource (National Academy of Sciences,Washington, DC, 1986). tion. Hundredsof other examplescould have illustratedthe same 5. MathematicalSciences: Some Research Trends (National Academyof Sciences, points; those chosen here are neither the deepest nor the most Washington,DC, 1988). 6. G. Kolata,Science 225, 156 (1984). important.They do suggest, however, the variety and scope of 7. T. Tymocszko,J. Philos.76, 57 (1979). today'smathematics. 8. E. Kitcher,The Nature of Mathematical Knowled,ge (Oxford Univ. Press,New York, Mathematicsis often definedas the scienceof spaceand number, 1983). 9. L. de Branges,in Proceedingsof theInternational Congress of Mathematzcians, 1986, as the disciplinerooted in geometryand .Although the A. Gleason,Ed. (AmericanMathematical Society, Providence, RI, 1988), pp. 25- diversityof modern mathematicshas alwaysexceeded this defini- 42. tion, it was not until the recent resonance of computers and 10. W. H. Lenstra,Jr., ibid.,pp. 99-120. 11. S. Landau,Not. Am. Math. Soc.34, 3 (1987). mathematicsthat a more apt definitionbecame fully evident. 12. D. Harris,ibid. 33, 443 (1986). Mathematicsis the scienceof patterns.The mathematicianseeks 13. A. Schonhage,in Proceedingsof theInter¢ational Congress of Mathematicians) 1986, A. Gleason,Ed. (AmericanMathematical Society, Providence,RI, 1988), pp. patternsin number, in space, in science, in computers,and in 131-153. imagination.Mathematical theories explain the relationsamong 14. S. Smale,ibid., pp. 172-195. patterns;functions and maps, operatorsand morphismsbind one 15. J. Gleik,Chaos (Viking, New York,1987). 16. C. Grebogi,E. Ott, J. A. Yorke,Science 238, 632 (1987). type of patternto anotherto yield lastingmathematical structures. 17. S. K. Donaldson,in Proceedingsof the InternationalCongress of Mathnnaticians, Applicationsof mathematicsuse these patternsto "explain"and 1986, A. Gleason,Ed. (AmericanMathematical Society, Providence, RI, 1988), predict naturalphenomena that fit the patterns.Patterns suggest pp. 43-54. 18. M. La Brecque,Mosaic (Washington) 16 (no. 1) (spring1985). other patterns, often yielding patterns of patterns. In this way 19. ibid.17 (no. 4) (winter1986-1987); ibid. 18 (no. 1) (spring1987). mathematicsfollows its own logic, beginning with patternsfrom 20. I. Peterson,Sci. News 131, 283 (1987). 21. W. C. Rheinboldt,FutureDirections in ComputationalMathematics, AZonthms, and scienceand completing the portraitby addingall patterns that derive ScientiflcSofsare (Societyfor Industrialand AppliedMathematics, Philadelphia, from the initialones. 1985). To the extentthat mathematicsis the scienceof patterns,comput- 22. T. A. Heppenheimer,Mosaic (Washington) 16 (no. 3) (1986). 23. Not.Am. Math. Soc.33, 716 (1986). ers change not so much the nature of the disciplineas its scale: 24. M. M. Waldrop,Science 238, 602 (1987). computersare to mathematicswhat telescopesand microscopesare 25. E. Witten,in Proceedin

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