Mathematics and Society

310 Pages·2016·15.018 MB·English

by  Wolfgang Konig

Preface
Mathematicspossessesacentralimportanceinoursociety.Itshapesandinfluencesmanyar-
eas of our daily life, from education and culture via technology and industry to physics and
information science, and more. Most obvious and most closest examples to our experiences
comprisethedesignandtheevaluationofinsuranceproductsandfinancingschemes,mobile
wirelesscommunication,electronicdevicesforaglobalpositioningsystem,andmuchmore.In
manyoftheseexamples,thecontributionofmathematicstothedevelopmentofthefunction-
alitywasdecisive,andtherearebillionsofusers,buthardlyanybodynoticesthemathematics
involvedanywhereinthefinalproduct.Indeed,thisphenomenonseemstobeevencharacter-
istic for mathematics: its rôle in the development of a new technology or device is vital, but
cannot be seen in the product itself. This is even more distinctive in intangible concepts and
proceduresthathavebeenshapedbymathematics,likeencryptionproceduresthatmakepos-
sible online banking, e-mail and the internet, or finance products that are adapted to special
marketsituations.Mathematicsisimportantforthedevelopmentoftechnologyandindustry
formanycenturiesandwillcontinuesoforever,andreal-worldproblemsinspireandacceler-
atemathematicalresearchinmanyways.
Buttherearealsofurther,immaterial,impactsofmathematicsonthesociety.Forexample,
someexcitingdevelopmentsinmathematicsraisedinterestingphilosophicquestionsthatare
at times discussed in public, like the question about the validity of proofs: how well must a
proof be checked, such that one can agree that the assertion has been proved? This is even
more difficult to decide for computer-aided proofs, the existence of which has gained quite
somepopularity,atleastwithinthemathematicalcommunity(thinkofthefamousfour-colour
problem, which says that four colours suffice for colouring a map without giving the same
colour to any two neighbouring areas). Popularity even outside the community is enjoyed by
questions about the quest for solutions to famous open problems like the Clay Millennium
Problems (www.claymath.org/millennium-problems). Another much-discussed aspect is the
general rôle of mathematics in education, as it inspires many people, including pupils, as a
freetime-occupationandstimulatesthemtofurtherinvestigationoftheirown.
Whenwetalkaboutmathematicsinsociety,thenwemeanmathematicsthathasaninfluence
onthedailylifeofasubstantialpartofthepeople,ormathematicsthatisanintegralpartof
oursocietyinoneoranotherway.Furthermore,wemeansuchkindofmathematicsthatcan
bedoneonlybyaprofessionalmathematicalexpert.Wetalkaboutthesearchofnewconcepts,
methods and proofs on a level that necessitates investigation by mathematical researchers
(jointlywiththeeffortsofspecialistsoftheapplicationfield,ifnecessary).
As we said at the beginning, much mathematical research whose results are decisive for
the wealth of the society and for solutions to its problems is so well hidden that the public
vi PREFACE
does not know about it, often not even the scientifically interested layman. Likewise, several
application fields are generally known, but not the way in which mathematics makes an im-
portantcontribution.Therearemanyreasonsforthisunfortunatesituation.Oneisofcourse
thatanunderstandingoftheproblemrequiressomeleastlevelofpreparatorytraininginthe
application field, which the layman typically does not have. Another one, and may be the de-
cisive one, is that the mathematics involved is quite abstract and specialised and needs even
more training in mathematics. And a third main reason is the lack of expert people who are
willingtotaketimeforexplainingthingsinawaythatthescientificallyinterestedlaymancan
understand.
This volume is an attempt to rectify the situation. It tells success stories about some im-
portantexamplesofmathematicsinsociety:areal-worldproblemtobesolved,mathematical
difficulties that the problem poses, and the rôle that mathematics plays in the solution. All
these stories are written by eminent and experienced mathematicians, who shaped, and con-
tinuetoshape,theapplicationfieldthattheydescribe.Thegoalwastocoveragreatdiversity
ofapplicationareas,andtoexplainthingsonanon-technicallevel,suchthatthescientifically
interestedlaymancantakehomequitesomemessage.Certainlyalsoprofessionals,boththose
in mathematics and those in the application area, will greatly benefit. But, given that fifteen
(rather diverse!) subjects are highlighted in this collection, nobody can be an expert in all of
them,buteveryscientificallyinterestedwillfindsomethingthatsheorheisabletofollowand
toenjoy.
Itisalwaysdifficulttotalkaboutmathematicsanditsapplicationstoreal-worldproblems;
even more, if readers without specialised education are addressed. The matter is generally
difficult, the author’s and the reader’s background differ a lot, and also the readership is
rather diverse: from expert mathematicians to mathematicians that work in other areas, and
from experts on the application field to the layman who is just interested in science. All the
authorsofthisvolumehavetriedtheirbesttoaddressatleastmorethanoneofthesegroups,
andsomeofthemevenaimedatthelattertypeofreader,whichisdefinitelythemostdifficult
task.
Beingpassionatemathematicians,theauthorsofthiscollectioncertainlypresentalsoagreat
deal of mathematical material; how else can you demonstrate its necessity and its benefits
better?Itmaybetheaexcitingaspectofeachoftheessaysthattheauthors’personaltastes
shinethroughthechoicesofthesubjects,throughthewaysofpresentationandthroughthe
comments.Itisalwaysinterestingtolearnaboutthepersonalviewoftheexpertatthethings,
evenmoreastheeditorialboardencouragedthecontributorstorevealsomeofthat,andmany
oftheauthorsgratefullyusedtheopportunitytodoso.
What are these application fields that we decided to present here? It was our purpose to
present a most diverse collection of aspects and areas of mathematics that have shaped and
continuetoshapeoursociety.Hereisasurvey.
Let us begin with a fundamental question about the interaction between mathematics and
the society: the question about how a mathematical expert should write about his field for
the public, such that (s)he will be understood and reaches her/his readership – precisely the
situationinwhichalltheauthorsofthisbookare!Readtheadvicesandentertainingexamples
of how to do that and how one should not do that by George Szpiro! – We proceed with
moreabstractapproachestothequestionhowmathematicsentersandinfluencessociety.The
enormouscomputerpowerthatmankindisnowabletouseproducednewwaysofproduction
of new mathematics, namely by computer-assisted proofs and by automated verification of
PREFACE vii
existing,complexproofsandfalsificationofconjectures.DonaldBaileyandJonathanBorwein
present deep and recent thoughts about this new field called experimental mathematics, and
touch also some fundamental questions like the one about when a proof can be considered
valid.
One of the reasons that mathematics is esteemed so high in our society is presumably its
rôleintheeducationasasciencethatismostamenabletoteachingmethodologiesthatappeal
tothenaturalwishofhumanstofindoutbyowndoing.Theconscienceforthisinoursociety
is significantly increasing only for a few years, but it has lead to (and has been increased by)
thefoundationofsciencecenters,hands-onmuseums,orevenspecialmathematicsmuseums.
One of the most well-known educational centers of this kind (at least in Germany), the Math-
ematikum, was founded some ten years ago in Gießen by our author Albrecht Beutelspacher.
Inhiscontribution,hedescribesitsconceptandthereasonsforitssuccess.Furthermore,he
givesanaccountonthehighlyinterestingquestion,whetherornottheinstitutionofallthese
mathematicsmuseumswasworthwhileandhowtheychangedtheattitudeofthepopulation
towardsmathematics.
Let us proceed with application areas that have a notable random component. In finance,
some few years ago, a big global crisis moved the world, and still the unspoken question re-
mainsintheroom,whetherornotthecrisishappenedbecauseoftheworkofmathematicians
orinspiteof it.ReadWalterSchachermayer’sopinionaboutthat!Healsoreportsontheinter-
estinghistoryofthefirstapplicationsoftheBrownianmotiontofinancesomehundredyears
ago.–Oneoftheubiquitouspartsofappliedmathematicsisthefieldofstatistics,whosegoal
istheprobabilisticdescriptionofreal-worldphenomena.Intheresearchofcancer,nowadays
manyapproachesmakeuseofanenormouslyhugeamountofcomplexdata.Asolutiontothe
problemhowtoextractusefulinformationfromthesebigdataliesbynomeansinastrategy
ofbrute-forcecomputingwithstrongerandstrongercomputerpower,butinthedevelopment
ofmoreandmoresophisticatedstatisticalmethods,notablyhandlingthehigh-dimensionality
ofthestructureinthedata.AadvanderVaartandWesselvanWieringengivesomeinsightin
the mathematical aspects of these methods. – Another application field of statistics is filter-
ingtheorywithinengineering.Thestoryofthismethodanditshistoricalsuccessesistoldby
OferZeitouni.–Probabilityplaysalsoanimportantrôleinpopulationmodelswithapplication
in biology. Especially in recent years, new models have been introduced that take care of the
lateststateofunderstandingoftheevolutionofbiologicalpopulations.JeanBertoinpresents
a pedagogical summary of some of the models that mathematical biologists work on most
activelyatthemoment.
One of the areas with the oldest, most immediate, and most discussed connections with
mathematicsisphysics.AlreadytheancientGreeksdeeplythoughtaboutthelawsthatunderly
all the matter and the mathematical way to describe them. By example of the Second Law of
Thermodynamics and the concept of entropy, Jürg Fröhlich brings to the reader the spirit of
the old quest for the understanding of fundamental laws that control a great deal of actions
aroundus.
Anotherlargeandclassicalpartofmathematics,eventheoldest,butstillverymodernand,
onecansay,ubiquitousfieldisgeometry,whichismoreup-to-datethanonemightthink.Ac-
tually,inarchitectureitplaysnowadaysanequallyactiverôleasthousandsofyearsago.More
specifically, Hellmut Pottmann and Johannes Wallner concentrate on problems from discrete
differentialgeometrythatanarchitecthastosolveif(s)heplanstodesignapieceoffreeform
architecture. Many pictorial examples from the practice illustrate the interplay between the
viii PREFACE
mathematical theory and the intended form of the building. – While this article is about the
geometry of objects that humans want to shape on their own, Christiane Rousseau concen-
trates on those shapes that appear in natural morphologies, for example animals and plants,
and follow rules of geometry that lead to beautiful and functional solutions. She brings the
rulesunderlyingthenaturalshapingforcestothesurfaceandexplainswhysimilargeometric
forms appear in most diverse connections in the nature. Striking relations with fascinating
mathematicalobjectsliketheKochsnowflakeappearinanewcontext.
Let us come to applications of mathematics in industry. First of all, does “industrial math-
ematics” exist at all? One of the most experienced experts in this field, Helmut Neunzert,
raises this question and extends it to the question “industrial mathematics versus academic
mathematics”.Thisisonlyastartingpointforalarge-scalesurveyonthehistoryandthecur-
rent situation in the relationship of mathematics, as is carried out in applied research, and
industrial mathematics that is meant to solve explicit tasks, including a lot of philosophical
considerationsandpersonalstatements!–Telecommunicationisaconcreteindustrialfieldin
which mathematics has much to do and to say, and its contributions are enormously diverse
and versatile. Holger Boche and Ezra Tampubolon concentrate on a particular question that
is ubiquitous in the theory and praxis of data transmission and can be resolved only by an
ingenious use of highly developed mathematics: how can one handle the huge differences in
theamplitudeofthetransmissionofasignalbymeansofanorthogonaltransmissionscheme?
They demonstrate that this annoying problem can be settled by use of some parts of mathe-
maticsthatareconsideredtobequitepure,likeadditivecombinatorics,butalsomoreapplied
disciplineslikefunctionalandharmonicanalysis.
Informationsecurityposesseverechallengesforthesocietyofthefuture.Cryptographynow
providesramifiedtechniquestodealwiththesechallenges.Itconstitutesawidefieldwhichis
nowheavilybasedonmethodsfromavarietyofmathematicaldisciplines,notablycomplexity
theoryandnumbertheory.ClausDiemexplorestheseconnections.Also,thelimitationsofthe
mathematicalapproachtoreallifesecurityarecriticallyaddressed.
Afieldthatwouldbenotguessedasafieldofmathematicalapplicationisthefieldofpoli-
tics, more precisely, voting systems. Actually, there are hardly any two votes that are carried
out under precisely the same set of rules, and slight changes in the voting rules can lead to
surprisingchangesintheresult,andnotonlytheoretically.WernerKirschgivesaflavourofa
mathematicalconceptofvotingsystems,itsbenefits,theeffectsthatitcontainsandthecon-
clusionsthatonecandrawwithintheconcept.Theconsequencesofsometheoreticalresults
forthesocietycanbeprettyimmediate,asheillustratesbymeansofhistoricalexamples.
The reader might have noticed that a field that is generally thought to have high affinity
tomathematicshasnotyet(oronlyonce,seetheabovementionedcontributiontostatistics)
beenaddressed:handlingbigdata.Last,butnotleast,thereisalsooneessaydevotedtothis
important subject, in the connection of investigation of the climate and the weather. Here it
isnotpossibletomakeexperiments,onehastorelyonmathematicaldescriptionsandpredic-
tions of the reality. Obviously, huge amounts of data are available, but the biggest problems
come from the enormous span of scales of the meaning of the data. Jörn Behrens describes
theestablishedmathematicalmodelsandmethodsingeoscience;inparticulartherôleofthe
importantfieldofuncertainty:whatcanmathematicsdoifwenotevenknowtheprobability
distributionoftheunknownquantitiesinourequation?
This ends our small survey of the articles contained in this volume. Let me express my
sincere thanks to the inspiring support of my colleagues that formed a awesome editorial
PREFACE ix
board: Jochen Brüning, Hans Föllmer, Michael Hintermüller, Dietmar Hömberg, Rupert Klein,
GittaKutyniok,andKonradPolthier.Theirbroadexpertiseandoverviewhelpedalottoidentify
a good choice of areas that should be contained in such a collection and experts that should
be approached as authors for a book with such an intention. Let me also thank Claus Diem
for careful reading and numerous hints and proposals, which led to a substantially better
readabilityofseveralofthecontributions.
I am also very grateful to the European Mathematics Society Publishing House and its staff,
which was most helpful and flexible at every stage of the production. This book is part of
theofficialdocumentsforthe7thEuropeanCongressofMathematics(7ECM),thequadrennial
CongressoftheEuropeanMathematicalSociety,whichisorganisedatTechnischeUniversität
BerlinonJuly18–22,2016.
Berlin,May2016 WolfgangKönig

Contents
Preface v
Thetruth,thewholetruthandnothingbutthetruth:
Thechallengesofreportingonmathematics
GeorgeG.Szpiro 1
Experimentalmathematicsinthesocietyofthefuture
DavidH.BaileyandJonathanM.Borwein 7
Whatistheimpactofinteractivemathematicalexperiments?
AlbrechtBeutelspacher 27
Mathematicsandfinance
WalterSchachermayer 37
Statisticsinhighdimensions
AadvanderVaartandWesselvanWieringen 51
Filteringtheory:Mathematicsinengineering,fromGausstoparticlefilters
OferZeitouni 71
Mathematicalmodelsforpopulationdynamics:Randomnessversusdeterminism
JeanBertoin 81
Thequestforlawsandstructure
JürgFröhlich 101
Geometryandfreeformarchitecture
HelmutPottmannandJohannesWallner 131
Somegeometriestodescribenature
ChristianeRousseau 153
Mathematicsinindustry
HelmutNeunzert 167
Mathematicsofsignaldesignforcommunicationsystems
HolgerBocheandEzraTampubolon 185
xii CONTENTS
Cryptology:Methods,applicationsandchallenge
ClausDiem 221
AMathematicalviewonvotingandpower
WernerKirsch 251
Numericalmethodsandscientificcomputingforclimateandgeosciences
JörnBehrens 281
Authors 295
Index 297
The truth, the whole truth and nothing but the truth:
The challenges of reporting on mathematics
GeorgeG.Szpiro
I would like to address the role and responsibility of the journalist who writes about mathe-
matics,andalsosharesomepersonalexperienceswithyou.
Inrecentyears,theattitudeofresearchmathematicianstowardsthepopularizationoftheir
subject has undergone a change. While they previously jealously, and sometimes haughtily,
guardedtheirworknearlyasasecretscience,theyhavenowbecomemoreattunedtothede-
mandsofthegeneralpublic.Afterall,evenifmathematiciansrequirelessfundsthanscientists
workinginotherfields,theystilldependonthetaxpayers‘support.Hence,thegeneralpublic
deservestoknowwhatitisthatmathematiciansdo.AsJochenBrüning,oneoftheeditorsof
this collection, pointed out, the change in attitude is apparent, for example, in the Notices of
theAMS.Whileitusedtobearatherdrynewssheet,whichreportedmainlyonmeetingsand
job openings, it has become a highly interesting and entertaining source of information. But
incontrasttomathematiciansreportingenthusiasticallyabouttheirortheircolleagues‘work,
journalists who write for mainstream publications have an important task as interpreters of
theworks‘significance.Theymust,however,alwaysremainawarethattheyarenoexpertsin
thefieldstheywriteabout.Acertainmeasureofhumblenessisrequiredontheirpartandthey
must ask for help, even if their interlocutors may be unwilling, because they still prefer the
protectednessoftheirivorytower.
BeforeIcanevenbegintodiscusshowmathematicsshouldbereportedoninnewspapersor
magazines,Imustfirstaddressahurdlethatneedstobeovercome.Thishurdlepresentsitself
not in the form of reluctant readers, but rather in that of sceptical editors. When I proposed
a monthly column on mathematics to the Neue Zürcher Zeitung (NZZ) am Sonntag, over a
dozen years ago, I was unsure whether this idea would be accepted. I asked a prominent
American colleague for his advice. He (if I write “he”, I may also refer to a woman) answered
thatnewspapereditorsaregenerallyverycreativepeoplewhohaveturnedtowriting(instead
of, say, the sciences) because they were often at odds with mathematics in school. This was
thus the basis of their scepticism of everything mathematical. Now, I was fortunate and had
willingeditors.And,tomypleasure,itturnedoutthatmanymorereaderswereinterestedin
acolumnaboutmathematicsthaneitherIormyeditorshadexpected.
Thequestionthatamathematicsjournalistmustaskhimselfconcernsasimplebutpowerful
formula:thetruth,thewholetruthandnothingbutthetruth.
Must a journalist who writes about mathematics follow this principle? At first glance, the
question seems irrelevant, almost provocative. Of course the truth is a common good that
should be a guiding principle for everyone – not just journalists – in all areas of life. In po-
litical reporting, for example, it is undisputable that journalists are absolutely obligated to
2 GEORGEG.SZPIRO
thetruth,thewholetruthandnothingbutthetruth.WhenIwearmyotherhat,thatofafor-
eign correspondent for the NZZ, it goes without saying that I always make a strong effort to
rigorouslyobserveallthreepartsofthismaxim.
But is that which is self-evident for journalists in general also applicable to the field of
mathematics?
Before I try to answer this question, I will first pose another one. Assuming that the work
of mathematics journalists is different from that of other reporters, should they perhaps let
themselvesbeinspiredbyarticlesinmathematicsjournals?Inthiscase,theanswerisasimple
one. It is a clear No – the requirements of an article written for a daily newspaper or weekly
magazinearemuchdifferentfromthoseofanarticlewrittenforamathematicsjournal.
An academic author of a journal article expects that the reader will carefully work his way
through the article, even if this requires effort. The journalist, on the other hand, has to
makeiteasyforthereader.
Theacademicaimsatannouncingresultsthathavebeenproveninastrict,rigorousway;the
journalistwantstoinform,butalsotoamusethereader.
Ajournalarticlewillbecitedmanyyearsorevendecadeslater,andamistakeinanargument
may be discovered even long after the article was written. Newspaper articles are usually
forgottenafteraday,andalloftheirflawswiththem.
Scientists only submit a manuscript for publication when they are sure that they cannot
improve it further. Journalists usually have a publication date thrust upon them and are
underpressuretomeeteditorialdeadlines.
Usually, in a scientific journal, an author has as much space as needed, within reasonable
limits.Inthenewspaper,onlylimitedspaceisavailable.
Inthislimitedspace,anarticleonmathematicsshouldincludethefollowing:agoodtitle,an
explanationoftheproblem,thehistoryoftheproblem,thebackgroundofthemathematician
whopopularizedit,unsuccessfulattemptstosolvetheproblem,thepersonalityofthemath-
ematicianwhoproducedtheproof,theprocedureusedtoarriveatthesuccessfulprooforat
leasttheideabehindit,theimplicationofthetheoremandsomeapplications.
This all has to be done within the space of 600, 1200, or, under the best of circumstances,
2500 words. Thus we arrive at the question of whether, under these circumstances the jour-
nalistcan–orshould–allowhimselftobeledbytheprinciple“thetruth,thewholetruthand
nothingbutthetruth.”
Iwillmakeitclearfromthebeginning:asamathematicsjournalist,itisoftennotpossible
to completely live up to this standard; quite frequently, it is not even possible to come close.
In my newspaper articles about mathematics, I am myself guilty of transgressing against all
three parts of this principle to a certain degree. Hence, the mathematics journalist is caught
betweentwoworlds:theworldofeverydayreportingandtheworldofscientificjournals.
Letusbeginwiththeexpectationsofreaderswhoareexpertsinthefield.
Oneexample:IntheNZZ,Ioncewrotethat“theNavier-Stokesequationcanbeexactlysolved
onlyinspecialcases[...]althoughapproximationscanbecalculatedusingcomputermodels,
thenumericalmethodsareveryproblematic.”Thisledareadertosendalettertotheeditor:
This could have been written in the Bild-Zeitung [a German tabloid newspaper].
Everymathematicsstudentknowsthat,ingeneral,differentialequationsarenot
solvableinclosedformetc.,etc.