Explosion of Maths

explosionof

mathematics

5 ......................... Foreword Mireille Martin-Deschamps and Patrick Le Tallec

7 ......................... Predicting the weather Claude BasdevantForecasting the weather or the climate is not an easy matter. It requires modelling of numerous natu-ral phenomena and interaction between several sciences, ranging from mathematics to biology, via com-puter science, physics and chemistry.

11 ......................... What lies behind mobile phones Daniel KrobThe mobile telephone is now a relatively common object. Who hasn't ever seen or used one ? But fewhave given a thought to the science and technology behind it.

15 ........................... Encrypting and decrypting: secure communications Jean-Louis NicolasThe modern world, where telecommunications occupy a central place, cryptography has a major stakeIt has also became a complex science, which cannot do without high-level mathematicians.

19 ......................... Managing complexity Pierre PerrierWhether it concerns manoeuvring a plane, or the mechanical resistance of a complicated structure, ormanaging automobile traffic, progress in all these areas doesn't come from purely technological inven-tions alone. It also involves abstract research, like the mathematical theory of control.

23 ......................... The bellows theorem Étienne GhysA ruler, a pencil, cardboard, scissors and glue: one doesn't need more to give a mathematician pleasure,and present interesting problems whose study often turns out to be useful in other areas, in totallyunexpected ways.

28 ........................ Finding a cancer-causing gene Bernard PrumThe development of modern biology, and of molecular genetics in particular, requires new mathema-tical tools. Example of statistics and its role in finding a gene related to breast cancer.

32 ......................... Wavelets for compressing images Stéphane MallatWhether they are stored digitally in computer memories, or they travel over the Internet, images takeup a lot of space. Fortunately, it is possible to ``condense'' them without changing the quality!

36 ........................ Preventing waves from making noise Daniel BoucheHow to escape being detected by radar? What is the optimal shape of a sound-proof wall? Can oneimprove the quality of sonographic images? To get a satisfactory answer, these questions require a tho-rough theoretical analysis

41 ......................... When art rhymes with math Francine DelmerScientists are not the only ones to be inspired by mathematics. Many artists have drawn the subject ofsome of their works from it. The converse is also sometimes true, as in the case of perspective, whereart has led to geometrical theories.

47 ......................... From DNA to knot theory Nguyen Cam Chi et Hoang Ngoc MinhThe biological activity of a DNA molecule depends mainly on the way it is arranged in space and the way in which it is twisted - things which fall within the province of the mathematical theory of knots

Contents

51 ......................... The philosopher and the mathematician Pierre Cassou-NoguèsThroughout their history, mathematics and philosophy have had a close and enigmatic relationship. Itwould be necessary to go as far back as Plato in ancient Greece and Descartes at the dawn of moderntimes. Let us cite here two great figures of the 20th century, David Hilbert and Edmund Husserl

56 ......................... How to rationalise auction sales Jean-Jacques LaffontThanks to the Internet in particular, auctions have become widespread. Modelling these sales processesmakes it possible to determine their rules and the optimal strategies for using them

61 ............................. The econometrics of sellingwines and bonds Philippe Février and Michael VisserGreat wines and Treasury bonds are sold at auction. But which type of auction should one adopt ? Tofind that out, one has to supplement the general modelling of auctions by econometric studies

66 ......................... Puzzles for airline companies Jean-Christophe CulioliProblems of organisation and planning faced by an airline company are similar to those met with inother sectors of industry. Operations research, with which tens of thousands of mathematicians andengineers in the world concern themselves, tries to solve these problems as well as possible.

70 ......................... The geometry of 11 dimensionsto understand Genesis ? Maurice MashaalPhysicists have aspired for a long time to building a theory which would cover all the elementary par-ticles and all their interactions. Since about twenty years ago, they have a trail that is promising. But toexplore it, they must navigate in highly abstract spaces where even mathematicians had not yet ventu-red.

75 ......................... Internet: modelling its trafficfor managing it better François BaccelliSpecialists in communication networks try to understand the statistical properties of the data trafficwhich they have to route. The management and the development of these networks depend upon it.

80 ......................... Financial options pricing Elyès JouiniThe financial world fixes the price of options by means of formulas which have been obtained thanksto relatively recent workin mathematics. The search for better formulas continues... and small-time speculators are not the only ones who are interested!

84 ......................... Communicating without errors:error-correcting codes Gilles LachaudFor detecting and correcting the inevitable errors which creep in during digital information transmission, specialists in coding theory make use of abstract methods which arise from algebra or geometry.

88 ......................... Reconstruction of surfaces Jean-Daniel BoissonnatReconstructing a surface from the knowledge of only some of its points: a problem that one comesacross often, be it in geological exploration, in recording archaeological remains, or in medical or indus-trial imaging.

92 ......................... Mathematicians in Franceand in the world Jean-Pierre BourguignonAt the end of the 19th century, there were very few ``geometers'', as mathematicians were formerly cal-led. In one century, their numbers have augmented considerably. Today, they are facing profound changesin their discipline

Today we live in a paradoxical situation. Mathematics, as a tool for learning rig-orous and logical thinking, is irreplaceable; it helps to develop intuition, imagination,and a critical sense. It is also an international language and an important element ofour culture. Moreover, by its interactions with other sciences, mathematics is playingan increasing role in designing and developing objects for our daily life. But most peo-ple tend not to be aware of this, and for them mathematics has lost its meaning. It isnow sometimes fashionable, even among those in responsible positions, to boast atbeing “zero in math”, or to challenge its usefulness.

One can find explanations for this paradox and incomprehension, which are specificto mathematics. It is a discipline which draws on its links with other sciences and thereal world, but also enriches itself: its theories are not superseded; they build uponone another. Conversely, even if a large number of researchers in mathematics areinterested, above all, in the intellectual and even the aesthetic side of their discipline,sometimes applications emerge in unexpected ways. Thus, applications enrich research,but by themselves cannot direct it.

This subtle balance between internal and external factors of development must be pre-served at all costs. Trying to limit mathematical research to its potential applicationswould be tantamount to its disappearance. On the other hand, stressing, the study ofstructures and the internal dynamics of the discipline, as was done in French mathe-matics for several decades beginning in 1940, resulted in delaying the development ofapplied mathematics in France, in contrast to what happened at the same time in theUnited States and the Soviet Union. The factors leading to progress are very often atthe frontiers of the discipline.

We are delighted that today mathematics has restored, and sometimes created, stronglinks with other sciences and with many sectors of the economy. The line between puremathematics and applied mathematics has become blurred: the most fundamentalmathematics is being used to solve more and more difficult problems. Thus, fields suchas algebraic geometry and the theory of numbers have found unexpected applicationsin coding theory and in cryptography. In the same way, the strong links between math-ematics and finance have helped in the evaluation, and even in the creation, of increas-ingly complex financial products in response to the needs and demands of the eco-nomic players.

Foreword

6 L’explosion des mathématiques

However, there still remains much to to be done to change the image of mathe-matics, and to make the assets and attractions of mathematics and its applications morewell-known. The goal of this document is to make mathematics familiar in its variousaspects - scientific, technical, cultural, sociological; to underline the diversity and uni-versality of a discipline which has close ties with physics, chemistry, economics and biol-ogy, as well as with history, music and painting. Mathematics is everywhere. Withoutit there would be no computers, no software systems, no mobile phones, no designworkshops for car and aeronautical manufacturers, no satellite localisation systems, sig-nal processing, genome decoding, weather forecasting, cryptography, smart cards, orrobots.

Beyond its role as an academic science and a basic learning tool in schools, mathe-matics is omnipresent in today's society. It follows, accompanies, and sometimes pre-cedes the current scientific and technological advances, which rely on the latest resultsin contemporary fundamental research, as much as they benefit from the accumulateddiscoveries of the past. Lastly, the need for mathematics grows with the accelerationin technological creation and change. One cannot do without it when confronted withthe need to develop, control, or analyse increasingly complex systems.

This need has been understood in the United States, and the NSF (the NationalScience Foundation, the federal organisation in charge of distributing university researchgrants) decided in 2000 to substantially increase its financial support for mathematics.Our chance lies in the fact that in France the teaching of mathematics remains one ofthe best in the world, and that our scientists and engineers are highly regarded inter-nationally. The number of Fields medals, equivalent to the Nobel Prize which does notexist for mathematics, won by Frenchmen is testimony to this. Recently, on the occa-sion of the third European Congress of Mathematics, which was held in Barcelona inJuly 2000, five of the ten award winners were French mathematicians. Let us maintainthis level of excellence.

Mireille Martin-DeschampsPresident of the SMF from 1998 to 2001

Patrick Le TallecPresident of the SMAI from 1999 to 2001

Claude Basdevant

Forecasting the weather or the climate is not an easy matter.It requires modelling of numerous natural phenomena

and interaction between several sciences, ranging from mathematicsto biology, via computer science, physics and chemistry.

where does the weather bulletin that asmiling lady presents every night on the tel-evision come from ? No longer from frogs orthermometers, but rather from supercom-puters which process huge amounts of data,obtained mainly by satellites, together withthe laws of mechanics and physics, and alsomuch, often recent, mathematics.

For computers to make forecasts, it is firstnecessary to develop what is called a numer-ical weather prediction model. Schematically,a prediction model for the range of up toeight or ten days describes the state of theatmosphere by the values taken by meteor-ological parameters (wind speed, tempera-ture, moisture, pressure, clouds, etc) at thecentres of « boxes » which partition the vol-ume of the atmosphere. These boxes havesides about fifty kilometres long and heightbetween a few tens and a few hundreds ofmetres. This imaginary partitioning of theatmosphere into boxes is necessary becauseit is impossible to specify the parameters at

every point in the atmosphere (there are infi-nitely many of them!). In theory, the smaller(and therefore the more numerous) the boxesare, the more accurate the description of thestate of the atmosphere is, and the more accu-rate the forecasts will be. But the sides of theboxes cannot in practice be made smaller thanabout fifty kilometres. Below that, the powerof even the most powerful computers wouldbe insufficient. The forecast must be ready intime, well within 24 hours!

Starting from the assumed state of the

Artist's impression of the boxes used for calculations in a weather orclimate prediction model. (Illustration L. Fairhead LMD/CNRS).

Predicting the weather

atmosphere at the beginning of the predic-tion period, the numerical model computesthe ensuing future evolution on the basis ofthe laws of physics and dynamics. The com-putation is performed stepwise, with timestepsof a few minutes. That is the principle that liesbehind numerical weather prediction, a prin-ciple that had been known since the begin-ning of the 20th century, but had to wait untilthe advent of the first electronic computersin the 1940's and 1950's before it could bepractically implemented.

Meteorological measurementscannot be used directly

The first problem that arises in the idealforecasting scheme that has been justdescribed is the definition of the`ìnitial stateof the atmosphere”. Actual observations arenot well-suited for that purpose. Surfaceweather stations are irregularly distributedover the globe and provide very few meas-urements in altitude. As for satellites, most ofthem sweep the Earth continuously, so theirmeasurements are not obtained at the sametime at all points. Moreover, satellites meas-ure quantities that are integrals over the depthof the atmosphere (in general they measurethe radiative energy flux over a given wave-length range) and not the meteorologicalparameters (wind, temperature, moisture, etc.)that enter the equations of the model.

One therefore has to deal with a hetero-geneous mass of data, irregularly distributedover the surface of the Earth, spread out over24 hours, from which to ̀ ìnitialise” a forecast,i.e., to construct the starting point of the pre-diction. However, thanks to the theory ofdynamical optimisation, a field to which the

Russian mathematician Lev Pontryagin (1908-1988) and the French mathematical schoolhave contributed much, methods known as``variational assimilation” could be developedin the 1980’s, which made possible an optimalreconstruction of the initial state of the atmos-phere. The basic idea underlying these meth-ods, which have been used operationally byMétéo-France since 2000, is to force the tra-jectory of the numerical model to pass ̀ `close”to the data observed during the previous 24hours. But variational assimilation is not theonly modern mathematical technique that hascontributed to deeply influence processing ofobservations: the use of neuromimetic net-works or of wavelets, invented less than twentyyears ago, has led to spectacular gains in effi-ciency, accuracy and speed in processing dataprovided by satellites.

Numerical analysisenters the picture...

Once the required initial state of theatmosphere is known, it is necessary to developthe computer programs which will calculate,on the basis of the physical laws, the weatherto come. The physical laws are built on a con-tinuous description of space and time. But ournumerical model handles only a finite, albeitlarge, number of boxes; similarly, there is atime interval of several minutes between twosuccessive computed states - one says that theproblem has been ̀ `discretised”. Transformingcontinuous physical laws into a discretised for-mulation, while preserving as much accuracyas possible, that is the object of numericalanalysis, a branch of mathematics which haswitnessed a real explosion since the advent ofelectronic computing. The aim of numericalanalysis is to solve equations to the very end,


i.e., to the determination of numerical values,while saving as much time and effort as pos-sible. Numerical analysis is necessary for thesimulation not to be a mere simulacrum, andfor evaluating the uncertainty of the fore-casts. For example, significant progress hasbeen made recently regarding methods forsimulating the transport of chemical speciesor particles by atmospheric turbulence. Thishas led to significant improvement in the studyand prediction of air pollution.

Can weather be predicted far inadvance? The theory of dynamicalsystems says `no'

We have talked of short-term weatherforecast, up to eight or ten days. But whydoes one not make long-term forecasts? TheAmerican meteorologist Edward N. Lorenz,in a famous article in 1963, showed that it isprobably hopeless to try. The atmosphere isa chaotic system, i.e., any error in the initial

atmospheric state, howeversmall, is quickly amplifiedover time, so quickly that aforecast beyond ten days iscompletely unreliable.Nevertheless, that does notmean that one cannot pre-dict climate - i.e., make a sta-tistical instead of a deter-ministic forecast, in order todetermine the average pre-cipitation or temperatureover a period of time, ratherthan the precise weather inBrittany on a particular dayin July. The stakes are high:our climate is threatened bygas emissions due to human

activities and it is necessary to predict thelong-term effect of the resulting perturba-tions. It is the theory of dynamical systemswhich provides the tools for climate model-ling.

This theory, of which the mathematicianHenri Poincaré was a great precursor at thebeginning of the 20th century, has under-gone significant progress in the last twentyyears. It makes it possible, for example, toidentify what mathematicians call attractors,and meteorologists weather regimes. It alsomakes it possible to determine which weatherregimes are most predictable and which aremost unstable. In the case of instability, anappropriate tool would be probabilistic cli-mate modelling, which explicitly takes intoaccount the randomness of the forecast.Probabilistic climate models, which are stilllittle developed, must be based on the newtools of the theory of stochastic partial dif-ferential equations and of statistics.

Ozone plume over the Paris area at an altitude of 300 m on August 7, 1998, at 4 PM.Colour coded, concentrations as simulated by the CHIMERE numerical model of LMD/IPSL;the measurements taken by an airplane are shown in the small boxes(Illustration MERLINof Météo-France).

Predicting the weather 9

From weather prediction to climateprediction

Climate prediction models closely resem-ble weather prediction models, with two fun-damental differences. They use larger ̀ `boxes”(with sides of 200 to 300 km); as the time overwhich the simulation is to be performed variesfrom a few months to hundreds, or even thou-sands, of years, the numerical cost a higherresolution would be prohibitive. But the mostsignificant difference comes from the factthat, as climate variations occur over longperiods of time, it is no longer possible to neg-lect the interactions between the atmosphere,oceans, ice, and even the biosphere. That iswhy a climate model must combine a modelof the atmosphere, a model of the oceans, asea-ice model, and a model of the biosphere.Beyond the computational complexity of sucha system, delicate mathematical problemsarise in the combination of these differentmodels, and in the specification of the con-ditions at the various interfaces betweenatmosphere and ocean, ocean and ice, etc.Also, for the calculations performed on ̀ `largeboxes” to remain meaningful, it is necessaryto evaluate the statistical effect, at the scaleof those boxes, of phenomena which occuron a much smaller scale (for example, whatis the statistical effect, on the energy budgetof a 300 km-wide box, of small cumulus clouds,with a size of a few km in diameter, thatdevelop within the box?). In all these ques-tions, there is scope for numerous futuremathematical developments.

Claude BasdevantLaboratoire de météorologie dynamique,

École normale supérieure, Paris etLaboratoire Analyse, géométrie et applications,

Université Paris-Nord.


A few references:

• Numerous references on “Numerical WeatherPrediction” can be found on the net.

• La Météorologie, issue no 30, special issue onnumerical weather prediction (2000).

• R. Temam et S. Wang, « Mathematical Problemsin Meteorology and Oceanography », Bull. Amer.Meteor. Soc., 81, pp. 319-321 (2000).

Daniel Krob

The mobile telephone is now a relatively common object.Who hasn't ever seen or used one ? But few have given

a thought to the science and technology behind it.

What liesbehind mobile phones

mobile phones are now commonly usedin many countries. But not so long ago the sit-uation was quite different. In 1985, there werea great number of wireless telephone systems,which were conceived, developed and mar-keted by large national operators, but whichwere mutually incompatible. Being differentin their design features, it was not possible tocommunicate from one network to another.To make them compatible, it was necessary toagree on a whole set of technical specifica-tions, i.e. on a common standard. This processbegan in the following five years, when theGSM (Global System for Mobile communica-tions) standard emerged in Europe, on the ini-tiative of France Télécom and DeutscheTelekom, the French and the German tele-phone operators at the time. The first com-mercial systems based on this standard cameinto existence at the beginning of the 1990’s.But it was only towards the middle, actuallycloser to the end, of that decade that the GSMemerged as the only real international stan-dard for mobile telephony. The current devel-opment of mobile networks of the third gen-

An X-ray of a mobile phone. The electronics of this instrument lookscomplicated, but it does not hint at the amount of mathematicalwork which was necessary to develop mobile telephone communica-tion. (Photo Stock Image)

eration is, in fact, a testimony to the impor-tance of GSM, insofar as the new third gen-eration standard, the UMTS (Universal MobileTelecommunications System), is a naturalextension of the GSM standard.

The GSM Standard is of great scien-tific and technological complexity

The user is hardly ever aware that behindthe mobile networks there is a great scientificand technological complexity. For example,the GSM standard represents more than 5000pages of technical specifications, difficult toread even for a specialist! And the GSM stan-dard is far from being fixed: an enormouseffort in research and development is beingmade, both by the large radiotelephonic engi-neering firms and by university laboratories,to continuously improve the quality and theeffectiveness of mobile networks. The GSM

standard is based on a whole set of elaboratetechniques coming from traditional telecom-munications engineering, from computer sci-ence, from mathematics and from signal pro-cessing. In particular, mathematics andcomputer science play a fundamental role inthe design and the correct operation of theinternal mechanisms of mobile networks.Mathematics provides the theoretical under-pinnings for almost all the basic steps involvedin the transmission of a telephone call origi-nating from a mobile phone. Computer sci-ence allows the transformation of these basicresults into efficient and effective protocols,which can be implemented in a mobile radionetwork.


A relay antenna for GSM mobile telephone communication on a farm in the countryside. (Photo REA)

Algorithms to digitize information,to chop it up into packets,to encrypt it, etc.

To illustrate the impact of these two dis-ciplines in mobile telephone communication,let us take a closer look at the way in whicha telephone call is handled after a user callsa number from his handset. First of all, thedata transmitted within a mobile network isentirely digital: it consist of ``packets”, i.e.,sequences of 0 and 1 of fixed length, emittedevery quarter of a second, which contain allthe information (speech, identification of themobile, quality of reception as measured bythe mobile, etc.) about a given telephone call.In addition to keeping track of the where-abouts of the users, the great differencebetween mobile and traditional fixed-line tele-phone communication lies of course in thefact that the packets of digital informationare transmitted by Hertzian waves and not bycable; this required the development of awhole host of specific algorithmic and math-ematical techniques. These involve distributedalgorithms, combinatorial optimization, dig-ital signal processing, algorithmic geometryand error-correcting codes, to name only somefields.

The information packets are not trans-mitted as they are. To ensure the confiden-tiality of communications, each packet isencrypted using a cryptographic protocol spec-ified by the standard and using secret keysspecific to each operator (and it is known thatcryptographic methods often rest on very elab-orate algebraic or geometric techniques andconcepts). Hertzian transmission also requiresa preliminary processing of each informationpacket. In fact, the Hertzian channel is sub-ject to several types of noise, which affect the

signals emitted by a mobile. For example,absorption and reflection of Hertzian wavesby buildings leads to an attenuation and adephasing of the signal emitted by a mobile.In the same way, each signal generates manyechoes, which should also be taken intoaccount. Consequently, a part of each packetof information is especially designed to recoverthe original signal from within the sea ofechoes in which it is immersed. These prob-lems have, of course, been studied for a longtime, both at the theoretical, as well as thepractical level. Nevertheless, it was necessaryto develop and to adapt a significant part ofthe mathematical apparatus classically usedin these contexts to meet the engineering con-straints specific to mobile networks.

Using graph theoryto allocate frequencies

The contribution of computer science andmathematics is not limited to processing dig-ital information as outlined (very briefly) byus. Algorithmic techniques in particular areessential in effectively managing the radiofrequencies available to each operator. Theauthorities rent out, at a rather expensive rate,frequency bands which each operator can use;however, only a small number, about 300, ofthe frequencies within this band can really beused. Two calls made at the same time fromtwo different mobiles, which happen to begeographically close, cannot be transmittedat frequencies close to each other withoutinterferences which affect the quality of thetransmissions. So it is necessary to know howto distribute, in an optimal way, the availablefrequencies among the users who far out-number the available frequencies. A humanbeing is incapable of precisely solving this type

What lies behind mobile phones 13


of problem in a reasonable amount of time.Algorithmic methods based on mathematicalmodels such as graph theory were instrumentalin designing the software, which makes it pos-sible to solve these problems of frequency allo-cation in an approximate way. All these prob-lems are very important from an industrialpoint of view, and are still the object of veryactive research.

Daniel KrobDirecteur de recherches au CNRS

(LIX, École Polytechnique)Responsable de la chaire

Ingenierie des systèmes complexes

Some references:

• D. Krob et E.A. Vassilieva, « Performance eva-luation of demodulation methods : a combina-torial approach », Proceedings of DM-CCG,Discrete Mathematics and Theoretical ComputerScience, pp. 203-214 (2001) (disponible enligne : http://dmtcs.loria.fr).

• X. Lagrange, P. Godlewski, S. Tabbane, RéseauxGSM-DCS (Hermès, 1997).

• J. G. Proakis, Digital communications (McGraw-Hill, 3e édition, 1995).

• C. Servin, Télécoms : de la transmission à l’archi-tecture de réseaux (Masson, 1998).

Jean-Louis Nicolas

The modern world, where telecommunications occupya central place, cryptography has a major stake.

It has also became a complex science, which cannot dowithout high-level mathematicians.

in March 2000, a head-line made it to the frontpages: ``Credit CardSecurity Compromised”.What had happened? InFrance, the secrecy ofcredit card transactionswas protected since 1985by using a method ofencryption involving alarge number N consistingof 97 digits. This numberN is the product of twolarge prime numbers, i.e.,of numbers which, like 7or 19, are divisible only by1 and by themselves. Thesecrecy of a credit cardtransaction consists pre-cisely in this pair of prime numbers; to calcu-late them starting from N was practicallyimpossible in the 1980’s. But with increasingcomputational power and improved mathe-matical methods, the size of the numbers N

which can now be factored in a reasonableamount of time has crossed a hundred digitsin the final years of the last century (the cur-rent record of 158 digits dates from January2002).

Encrypting anddecrypting: securecommunications

Paying with a credit card or purchasing on the Internet: cryptographic methods involving beauti-ful mathematics are essential for the security of these operations. (Photograph: Getty Images.)

An astute computer scientist, Serge Humpich,was able to find the two ultra-secret primefactors, whose product is N, and had used themto fabricate counterfeit credit cards. Inresponse, to guarantee the security of our lit-tle plastic rectangles, those charged with issu-ing credit cards came up with a new numberN, which was considerably bigger.

Modern cryptography,at the crossroads of mathematicsand computer science

This adventure illustrates the considerableimportance that the science of encryption, i.e.,of coding messages in order to make themillegible by indiscreet people, has acquiredtoday. Encrypting and deciphering secret mes-sages is an activity which is several centuries,even thousands of years, old. This activity hasnow largely surpassed its diplomatic or mili-tary applications to enter into whole new areasof civil communications: authentificationschemes, banking transactions, electronic com-merce, protection of websites and computerfiles, etc.

Cryptography has undergone majoradvances in the last few decades. It has there-fore became a complex science, where progressis generally made by specialists who havereceived advanced training in mathematicsand in computer science. This specialisationemerged during the Second World War. Wenow know that the Allies' ability to deciphermessages coded by the famous GermanEnigma machines played a decisive role dur-ing the war.

Moreover, it was the eminent British math-ematician, Alan Turing, one of the fathers oftheoretical computer science, who made an

essential contribution to deciphering thosemessages.

In the 1970’s, cryptography underwent asmall revolution: the invention of ̀ `public-key”cryptography in the form of the RSA method.What is it all about? Up until then, those whowanted to exchange secret messages had tohave a secret key in common, and there wasa great risk of the enemy intercepting this key.The RSA protocol, named after its three inven-tors (Ronald Rivest, Adi Shamir and LeonardAdleman), overcame this difficulty. The methoduses two keys: a public key - known to all - forencrypting, and a key for decrypting, whichremains secret. It is based on the principle (usedlater to protect credit card transactions, as wehave seen above) that it is possible to con-struct some very large prime numbers (of ahundred, even a thousand digits or more), butthat it is extremely difficult to find the primefactors p and q of a large number N = p x qwhen one knows only the product N. Roughlyspeaking, the knowledge of N amounts toknowing the public key to be used for encryp-tion, whereas the knowledge of p and qamounts to knowing the secret key to be usedfor decrypting.

Obviously, if someone finds a method forquickly factoring large numbers into theirprime factors, the RSA protocol would becomeuseless. But it is also possible that mathe-maticians will one day prove that such amethod does not exist, which would reinforcethe safety of the RSA protocol. These are fun-damental research problems.

Methods like the RSA protocol, which useadvanced number theory, teach us an impor-tant lesson: mathematical research (researchon prime numbers, in particular) which may


seem to be completely unrelated to practicalmatters, may turn out to be crucial for someapplication many years, or decades later, in acompletely unpredictable way. In his book Amathematician's apology, the great Britishanalyst G. H. Hardy (1877-1947), who was afervent pacifist, took immense pride in work-ing in number theory, an absolutely pure field,and at never having done anything whichcould be considered `ùseful”. It was perhaps`ùseless” at the time. That is no longer thecase today.

Elliptic curves: algebraic geometryat the service of secretagents

And it is not only number theory. Otherfields of mathematics, which were oncethought to have no applications, contributeto the science of encryption. In the last fewyears promising cryptographic methods basedon principles close to those of the RSA pro-tocol have appeared - for example, themethod called discrete logarithm. This in turnled to methods which are based on proper-ties of elliptic curves. These are not curves hav-ing the shape of an ellipse, but curves whosestudy began in the 19th century to solve thedifficult problem of calculating the perime-ter of an ellipse. The coordinates (x, y) of pointson these curves satisfy an equation of the typey2 = x3 + ax + b, and they have interesting prop-erties studied in algebraic geometry, a vastfield of contemporary mathematics. For exam-ple, using a suitable geometrical construction,it is possible to define a law of addition onthe points of an elliptic curve. More gener-ally, geometrical objects like elliptic curveshave many arithmetic properties - which onecontinues to explore - that are likely to be use-

ful in cryptography. This was how a crypto-graphic method called elliptic curve discretelogarithm was developed.

Recently, another direction has been dis-covered. At the International Congress ofMathematicians held in Berlin in 1998, PeterShor, of the AT & T Labs, obtained theNevanlinna Prize for his work on quantumcryptography. What does this term mean?Some years ago, physicists and mathemati-cians theorised that it would one day be pos-sible to make a quantum computer, i.e., onewhich would exploit the bizarre laws of quan-tum physics to function, laws which govern

Encrypting and decrypting... 17

Graph of the elliptic curve given by the equation y2 = x3 + 1. Ellipticcurves have a remarkable property: one can “add” their points accord-ing to the procedure represented in the drawing. The “addition” thusdefined satisfies the usual arithmetic laws, such as (P1+P2) + P3=P1+ (P2+P3). Certain methods of modern cryptography make use of ellip-tic curves and their algebraic properties.


the world of the infinitesimally small. It wasproved that such a computer, if it could bebuilt, would be able to factor large numbersvery quickly and would thus make the RSAmethod ineffective. In fact, research pertain-ing to the concrete realisation of a quantumcomputer was published very recently, in theBritish journal Nature (cf. the last referencebelow). On the an other hand, scientist haveworked out quantum cryptography protocols,i.e., methods of encryption using objects (pho-tons, atoms...) obeying the laws of quantumphysics. These quantum protocols could guar-antee absolute security. All these approachesare being studied and may be put in practicein a few years...

Jean-Louis NicolasInstitut Girard Desargues, Mathématiques,

Université Claude-Bernard (Lyon 1)

Some references:

• D. Kahn, The Codebreakers: The ComprehensiveHistory of Secret Communication from AncientTime to the Internet (1996, Revised edition).

• S. Singh, The Secret History of Codes and CodeBreaking (Fourth Estate; New Ed edition 2000).

• P. Ribenboim, The New Book of Prime Number Records, (3rd edition, Springer, 1991).

• D. Stinson, Cryptography, Theory and Practice(3rd edition, 2005, CRC Press Inc).

• L. M. K. Vandersypen et al., « Experimental realiza-tion of Shor’s quantum factoring algorithm usingnuclear magnetic resonance », Nature, vol. 414, pp.883-887 (20 décembre 2001).

Pierre Perrier

Whether it concerns manoeuvring a plane, or the mechanical resistanceof a complicated structure, or managing automobile traffic,

progress in all these areas doesn't come from purelytechnological inventions alone. It also involves abstract research,

like the mathematical theory of control.

it is easy to understand the importance ofknowing how to control the reaction of aplane or a rocket to atmospheric turbulences,the steps to follow in the case of an accidentin a nuclear powerplant, to manage thedistribution in anelectric network incase of a breakdown,etc. Under normal cir-cumstances the pur-pose of controllingthings is to optimisesomething, or improv-ing the performance,or economise in mate-rials or expense: suchis the case when onewants to maintain asatellite in its properorbit while using theminimum amount offuel.

Let us look at the example about manag-ing the distribution in an electric network inthe event of a breakdown. An accident suchas a short circuit or a disconnection(caused for

Managing complexity

The Vasco de Gama bridge on the Tage in Lisbon. The resistance of a complex structure such as a bridgecan be controlled in an active way by placing, in some well-selected places, devices which modify its mechan-ical characteristics in response to the movements of the structure, in order to counteract the effects of reso-nance. The mathematical theory of control deals with such situations. (Photo Gamma/Gilles Bassignac)

example by a fallen electricity pole), or over-consumption of energy in a given place, canhave a cascade effect on the network.However, it is generally impossible to makean exhaustive study of all possible accidents,or to calculate exactly each step in the prop-agation of the effects of each possible acci-dent. The number of possibilities is enormous,in any case much too large for even the mostpowerful computers.

One must instead design a simplified math-ematical model describing the network andits operation. With the help of tests and cal-culations of reasonable size, such a modelmakes it possible to determine, at least roughly,the behaviour of the system. In return, thatcan help to improve the design of the net-work. But one would also like to control a crit-ical situation caused, for example, by an over-load localised in one place, or spread out overa whole area. In other words, one would liketo know which sequence of actions the com-mand unit must carry out in order to minimisethe consequences of the breakdown. Is it, inprinciple, possible to obtain such knowledge?Do there exist optimal strategies of control?If so, which ones? And finally, which algorithmscan be employed to check them by numericalsimulation on a computer, before trying themout in real life?

It is important to study this problem in aframework of resource management if onedoesn't want to waste energy, or be the vic-tim of power everywhere. This is our first exam-ple of a complex control problem where math-ematicians - with help from mathematicallogic, the theory of the numbers, probabilitytheory, analysis and control theory - make theircontribution. At the very least, they can pro-vide some certainty about the existence of an

acceptable solution and the means of obtain-ing it - a solution that will have to be later val-idated by experiment.

Preventing bridges from collapsing

Complexity does not lie in networks alone.It can reside in the way in which an object, likea bridge, reacts. The behaviour of such a struc-ture depends on a great number of parame-ters, for example, its response to vibrations,among other things. As we all know, the vibra-tions of a bridge can be caused by the passageof a row of trucks, or by the wind during astorm. Sometimes, this phenomenon getsamplified to the extent of breaking the bridge.A bridge, like any other mechanical structure,has a series of characteristic vibrational fre-quencies; if the external disturbance feedsenergy to these characteristic vibrational fre-quencies, a phenomenon called resonanceoccurs and the bridge accumulates energy inits characteristic modes of vibration. Thus thesevibrations get amplified as long as the exter-nal disturbance lasts, or as long as the struc-ture can resist the resulting mechanical stress.

To control such phenomena it is necessaryto understand them, to know how to predictthem and to set up technical devices capable ofthwarting these dangerous resonances. It iscalled passive control when one calculates thepoints where it is necessary to install shockabsorbers, which will absorb enough energybefore it accumulated in critical places. But it iscalled active control if, once these critical pointsare located, one installs, in some well-chosenplaces, an active device, an actuator, which actsaccording to the amplitude of displacement ofthe critical points, in order to prevent the struc-ture from evolving in any dangerous direction.


It is the mathematical analysis of the system inquestion which determines the appropriate sitesfor the sensors and the actuators, and the mostsuitable control procedures.

Unfortunately, in general it is impossible tocalculate exactly the behaviour of a system inthe absence of control, or its sensitivity and abil-ity to be controlled. Normally the reason is eitherthe mathematical complexity of the problem ifit is nonlinear (the impossibility of breaking itup into a sum of simple elements, which can beconsidered almost independent from the math-ematical point of view), or that it would taketoo long to make the computation on anymachine. Consequently, control is often imper-fect. It may happen, for example, that we suc-ceed in controlling some vibrational mode onlytemporarily - initially the external energy getsaccumulated in a number of low-amplitudevibrational modes, before combining with itselfto re-appear in a smaller number of high-ampli-tute modes. Much remains to be done to bet-ter understand these processes and remedy theirnegative effects

Withstanding turbulences

Let us look at a third example: fluid flow-ing at high speed, like the air flowing aroundan airplane or a rocket at takeoff, or wateraround a fast boat. In these situations one isconfronted with turbulence, i.e., with com-plex and unstable movements of the fluid,with a perpetual destruction and rebuildingof structures so complicated that they appearto be completely random or disorderly.Turbulences can considerably obstruct themovement of a vehicle in the sky, in water, oron the ground. Obviously, control is muchmore difficult to achieve here. But these prob-

lems have a great practical significance.Engineers have tried, by trial-and-error, andby being inspired, for example, by the flightof birds, to design planes so as to ensure a cer-tain controllability of the flow. They have par-tially succeeded in this by reinforcing the sur-face of the wings at the leading and thetrailing edges, by placing sensors in placeswhich are less affected by the disturbance,and actuators - guiding surfaces - at the crit-ical locations, close to the trailing edges.

The mathematical theory of control ini-tially helped to find these empirical resultsand then to propose action strategies or design

plans which reinforce or decrease, accordingto what is desired, the sensitivity to actions ofthe human operator or the external distur-bances. Now we are at a stage where we canidentify elementary active control devices,which would act on an almost microscopicscale, like that of a layer of fluid of some tenthof a millimetre in thickness: for example, atthe level of small shutters or micromechanismsallowing one to locally deform the profile ofthe vehicle at the critical points of fluid flow.

Managing complexity 21

The top image shows a relatively regular flow of supersonic fluid. Inthe bottom image, the action of a small jet of fluid injected laterallyhas the effect of leading to instabilities in the flow. Such an experimentillustrates the idea that one can modify a flow using small devices, inparticular with a view to controlling it. (Photo Erwan Collin-LEA/CEAT-University of Poitiers).


By coordinating the actions of a large num-ber of micro-devices of this kind, one obtains,on the macroscopic scale, a fluid flow havingthe desired properties. In the field of con-trolling the turbulence of fluids, mathemati-cal research together with physical and tech-nical tests will open a world of performanceunimaginable a few years ago; a world where,in order to achieve the same effect, the energyor the size of the necessary devices will havedecreased by more than an order of magni-tude.

Control theory brings into play vari-ous mathematical fields, in particularthe theory of the differential equations

The problems of control, which we havedescribed here, may deal as much with theordinary windscreen wiper of a car as with themost elaborate rocket launcher. The theory ofcontrol, born in the 1940’s and 1950’s mainlyin relation to aerospace activities, draws itsmethods and its concepts from several branchesof mathematics. Mainly it concerns differen-tial equations (where the unknown is a func-tion) and partial differential equations (dif-ferential equations where the unknownfunction is a function of several variables), anold and vast field of study, which is neverthe-less still very active. Indeed, the behaviour ofthe majority of systems that we come acrossin the real world can be modelled using suchan equation. A problem of control gets trans-lated into one or more differential equations,or partial differential equations, which con-tain terms representing control actions pre-scribed by the designers. Let us denote theseterms of control by C and by f, the functionrepresenting the behaviour of the system; f isthe solution of differential equations in which

C appears, and therefore f depends on C. Thegoal of the theory of control is then, roughlyspeaking, to determine the appropriate C suchthat f, the behaviour of the system, is accept-able. For the mathematician, it is not so mucha question of doing it for such and such par-ticular equations, but rather to obtain generalresults which are valid for many classes of equa-tions and thus applicable to many differentsituations.

In France, the theory of control occupiesa place of honour in the brilliant school ofapplied mathematics created by Jacques-LouisLions (1928-2001). But a good school of math-ematics does not suffice in itself. It is also nec-essary that its results are known and appliedby all those who need them. Therein lies theimportance of reinforcing the links betweenthe mathematical community and engineers,chemists, or biologists.

Pierre PerrierAcadémie des sciences and

Académie des technologies, Paris.

Some references:

• J. R. Leigh, Control theory. A guided tour (PeterPeregrimus, Londres, 1992).

• J. Zabczyk, Mathematical control theory: an intro-duction (Birkhaüser, 1992).

• J.-L. Lions, Contrôlabilité exacte, perturbations etstabilisation de systèmes distribués (Masson, 1988).

The bellows theoremÉtienne Ghys

A ruler, a pencil, cardboard, scissors and glue:one doesn't need more to give a mathematician pleasure,

and present interesting problems whose study often turns outto be useful in other areas, in totally unexpected ways.

let us build a card-board pyramid... Onestarts by cutting outthe design SABCDE ina sheet of cardboardas indicated in figure1, then one foldsalong the dottedlines and, finally, oneglues the sides ASand ES.

The result is a kindof cone whose vertex is the point and whosebase is the quadrilateral ABCD. This object isflexible. If held in the hand, the quadrilateralABCD can be deformed and opened or closeda little: the construction is not very solid. Tocomplete the pyramid we need to cut out asquare from the cardboard and to stick it ontothe quadrilateral to form the base. After thisoperation the pyramid is sturdy and rigid. Ifone puts it on a table it does not collapse. Ifone takes it in hand and tries to deform it

(softly!), one is unable to do it without deform-ing the cardboard face.

In the same way, a cardboard cube is rigid,as everyone must have observed at one time.What about a more general polyhedron, hav-ing perhaps thousands of faces? Is the Géode,a dome at La Villette in Paris, rigid? This lastquestion suggests that the subject of rigidityor flexibility is perhaps not only a theoreticalone!

Figure 1. The construction of a pyramid from a piece of cardboard. If the base ABCDA is removed, theobject is flexible.

A problem going back to Antiquity,but which is still relevant

The problem of rigidity of these types ofobjects is very old. Euclid probably was awareof it. The great French mathematician Adrien-Marie Legendre became interested in ittowards the end of the 18th century andtalked to his colleague Joseph-Louis Lagrangeabout it, who in turn suggested it to theyoung Augustin-Louis Cauchy in 1813. It wasto be the first major result of baron A.-L.Cauchy, who went on to become one of thegreatest mathematicians of his century

Cauchy was interested in convex poly-hedra, i.e., polyhedra which do not have anyinward-pointing edges. For example, thepyramid that we built or the surface of afootball are convex, while the object drawn

on the right of figure 2 is not.

The theorem established by Cauchy is thefollowing: any convex polyhedron is rigid.That means that if one builds a convex poly-hedron with indeformable polygons (madeof metal, for example) adjusted by hingesalong their edges, the overall geometry ofthe object prevents the play of joints. Thecone that we built is flexible, but that doesnot contradict the theorem: a face is miss-ing, and it is the last face which makes thepyramid rigid...

Doing mathematics means proving whatone claims! It so happens that Cauchy's proofis superb (even if it was pointed out later thatit is incomplete). There is unfortunately no ques-tion of giving an idea of this proof in this shortarticle, but I would like to extract from it a``lemma”, i.e., a step in the proof.

Let us place on the ground a chain madeup of some metal bars joined at the ends, as infigure 3. At each angle of this polygonal line,let us move the two bars in order to decreasethe corresponding angle. Then the two endsof the chain come closer. Does that seem obvi-ous to you? Try to prove it...

For a long time many mathematicians won-dered whether nonconvex polyhedra were also


Augustin-Louis Cauchy (1789-1857), one of the great mathemati-cians of his time. (Photo Archives de l'École polytechnique)

Figure 2. A convex polyhedron and a star-shaped, non-convex, poly-hedron.

rigid. Can one find a proof ofthe rigidity which would notuse the assumption of con-vexity? Mathematicians likestatements in which all theassumptions are necessary toobtain the conclusion. Onehad to wait more than 160years to know the answer inthis particular case.

In 1977, the Canadianmathematician RobertConnelly created somethingsurprising. He built a (quitecomplicated) polyhedron,which is flexible, and, of course, nonconvexnot to contradict Cauchy! Since then the con-struction has been somewhat simplified, inparticular by Klaus Steffen. In figure 4, I havegiven a design which will allow the reader tobuild the ``flexidron” of Steffen. Cut it outand fold along the lines. The solid lines rep-resent edges pointing outward, and the bro-ken lines correspond to edges pointing inward.Stick the free edges in the obvious way. Youwill obtain some kind of creature and you willsee that it is indeed flexible (a little...).

Does the volume of a polyhedronchange when it is deformed?

At the time, mathematicians wereenchanted by this new object. A metal modelwas built and put in the tea room of theInstitut des Hautes Études Scientifiques, atBures-sur-Yvette, near Paris, and one couldhave fun making this thing move; to tell thetruth, it was not very pretty, and squeaked alittle. The story goes that Dennis Sullivan hadthe idea of blowing some cigarette smokeinto Connelly's flexidron and he noticed thatwhile the object moved, no smoke came out...So he got the idea that when the flexidron isdeformed, its volume does not vary! Is thisanecdote true? Whether true or not, Connellyand Sullivan conjectured that when a poly-hedron is deformed, its volume remains con-stant. It is not difficult to check this propertyin the particular case of the flexidron ofConnelly or for that of Steffen (through com-plicated and uninteresting calculations). Butthe conjecture in question considers all poly-hedra, including those which have never beenbuilt in practice! They called this question the

The bellows theorem 25

Figure 3. If one decreases the angles which the segments form witheach other, the ends of the chain of segments come closer.

The “géode de la Vilette” in the Cité Des Sciences in Paris, is a convex polyhedron of 1730 tri-angular faces. The rigidity of this structure of joined polyhedra gives rise to an interesting math-ematical problem which was solved only in 1997. (Photo Cosmos/R. Bergerot)

``the bellows conjecture”: the bellows at thecorner of the fireplace eject air when they arepressed; in other words, their volume decreases(besides, that is what they are meant for). Ofcourse, true bellows do not provide an answerto the problem of Connelly and Sullivan: theyare made of leather and their faces becomedeformed constantly, in contrast to our poly-hedra with rigid faces.

In 1997, Connelly and two other mathe-maticians, I Sabitov and A. Walz, finally suc-ceeded in proving this conjecture. Their proofis impressive, and once more illustrates theinteractions between different areas of math-ematics. In this eminently geometrical ques-tion, the authors have used very refined meth-ods of modern abstract algebra. It is not a proofthat Cauchy ``could have found”: the mathe-matical techniques of the time were insuffi-cient. I would like to recall a formula whichone used to learn at secondary school at onetime. If the sides of a triangle are a, b and cin length, one can easily calculate the area ofthe triangle. For that, one calculates first thesemi-perimeter p=(a+b+c)/2 and then oneobtains the area by extracting the square root

of p(p-a)(p-b)(p-c). This pretty formula bearsthe name of the Greek mathematician Heroand its origins are lost in antiquity. Can onecalculate, in a similar way, the volume of a poly-hedron if the lengths of its edges are given?Our three contemporary mathematicians haveshown that one can.

They start from a polyhedron built froma certain design having a certain number oftriangles, and they call l1, l2, l3, etc. the lengthsof the sides of these triangles (possibly verymany). They then find that the volume V ofthe polyhedron must satisfy an equation ofthe nth degree, i.e. an equation of the forma0 + a1V + a2V2+ ... + anVn = 0. The degree ndepends on the design used, and the coeffi-cients (a0, a1, etc.) of the equation dependexplicitly on the lengths l1, l2, l3, etc. of thesides. In other words, if the design and thelengths of the sides are known, the equationis known. If the reader remembers that anequation has in general one solution if it is ofthe first degree, two solutions if it is of thesecond degree, he will be able to guess thatan equation of degree n cannot have morethat n solutions. Conclusion: if one knows the

design and the lengths, one doesnot necessarily know the vol-ume, but it is at least known thatthis volume can take on only afinite number of values. Whenthe flexidron is deformed, its vol-ume cannot vary continuously(otherwise the volume wouldtake on an infinity of successivevalues); this volume is ̀ `blocked”and the bellows conjecture isestablished...


Figure 4. The model for the flexidron of Steffen.

Yes, the bellows problem is worthy ofinterest!

Is this problem useful, interesting? Whatis an interesting mathematical problem?That's a difficult question, which, of course,mathematicians have been contemplating fora long time. Here are some partial answers,some indicators of ``quality”. The history ofa problem is the first criterion: mathemati-cians are very sensitive to tradition, to prob-lems stated a long time ago, on which math-ematicians of several generations haveworked. A good problem must also be statedsimply, its solution must lead to surprisingdevelopments, if possible connecting very dif-ferent fields. From these points of view, theproblem of rigidity, which we have just dis-cussed, is interesting.

The question as to whether a good prob-lem must have useful practical applications ismore subtle. Mathematicians answer it in avariety of different ways. Undoubtedly, ̀ `prac-tical” questions, arising for example fromphysics, are very often used as a motivationfor mathematics. Sometimes it is a questionof solving a quite concrete problem, but therelationship is often less direct: the mathe-matician uses the concrete question only as asource of inspiration and the actual solutionof the initial problem is no longer the truemotivation. The problem of rigidity belongsto this last category. The physical origin israther clear: the stability and the rigidity ofstructures, for example metallic structures.For the moment, Connelly‚s examples are ofno use to engineers. However, it is clear thatthis kind of research will not fail, in an inde-terminate future, to provide a better overallunderstanding of the rigidity of vast struc-tures made up of a large number of individ-

ual elements (macromolecules, buildings, etc.).It is thus a purely theoretical ``disinterested”kind of research, but which has a good chanceone day of being useful ...

Étienne GhysÉcole Normale Supérieure de Lyon,

CNRS-UMR 5669

The bellows theorem 27

Some references:

• M. Berger, Geometry. I and II - Translated fromFrench by M. Cole and S. Levy, (Universitext.Springer-Verlag, Berlin, 1987).

• R. Connelly, I. Sabitov, A. Walz, « The bellowsconjecture », Beiträge Algebra Geom., 38 (1997),n° 1, pp. 1-10.

• R. Connelly, « A counterexample to the rigidityconjecture for polyhedra », Institut des HautesÉtudes Scientifiques, Publication Mathématiquen° 47 (1977), pp. 333-338.

• N. H. Kuiper, « Sphères polyédriques flexiblesdans E3, d’après Robert Connelly », SéminaireBourbaki, 30e année (1977/78), exposé n° 514,pp. 147-168 (Lecture Notes in Math. 710,Springer, 1979).

Bernard Prum

The development of modern biology, and of molecular geneticsin particular, requires new mathematical tools. Example of statistics

and its role in finding a gene related to breast cancer.

innumerable diseases have a hereditary com-ponent: the risk for an individual of gettingthe disease is higher if he is a carrier of a genefor susceptibility to this disease. That is whymodern genetics seeks to understand the roleof various genes, and in particular their rolein the etiology of diseases, in the hope of oneday finding a cure. Let us take the example ofbreast cancer, which, in France, affects approx-imately one woman in eight. Apart from var-ious risk factors (food, tobacco, exposure toradiation, etc.),a few years ago scientists iden-tified a gene whose mutations are present ina high percentage of women who get this can-cer. This gene was baptised BRCA1 (for breastcancer 1). Such a biomedical result could beobtained only by a succession of statisticalanalyses, which as we shall see made it possi-ble to locate the gene with greater and greaterprecision.

For a long time genetics was ignorant ofthe physical nature of genes. It is only in the

Findinga cancer-causing gene

In this false-colour mammography, a cancerous tumour is visible inpink. A part of the research on breast cancer is devoted to its geneticaspect. The theory of statistics plays an important role here. (PhotoKings College School/SPL/Cosmos)

last twenty years that one hashad access on a large scale tosequences of DNA, the molecu-lar chain which carries thegenetic information transmittedfrom parent to child.Nevertheless, ignorance of thechemical composition of genesby no means prevented preciseresults on the heredity of certaintraits.

The first question whichoccurs vis-à-vis a disease such asbreast cancer is: `ìs this a genetic disease, arethere genes which predispose the person tothis disease?” For cancer in general, the answerwas uncertain for a long time. One expects apositive answer if one comes across familieswith a higher incidence of the disease; if thedaughter or the sister of an affected womancarries a greater risk than the risk incurred bythe population at large. The basic data aboutsuch Œpedigrees, which have been availableto genetic statisticians for a long time, is rep-resented in figure 1.

What does such a pedigree tell us? Weknow, almost since Mendel, that a hereditaryfeature is often determined by a ̀ `gene”, whichcan take several forms, called its alleles. Eachindividual inherits an allele from the father andan allele from the mother; one of these twoalleles is then randomly passed on to each off-spring. When studying the transmission of a dis-ease, the geneticist proposes a model in whichcertain genes and alleles are incorporated. Thestatistician must then test this model using suit-able statistical tools, which will make it possi-ble, for example, to eliminate the simplesthypotheses, such as: ̀ `the disease being studieddoes not have any genetic component”.

In the case of diseases (such as breast can-cer) with complex etiology, which are beingstudied more and more, where environmentalfactors come into play or where incidencedepends upon age, one has to refer to the sta-tistics of processes. This is an elaborate branchof mathematics, which is based mainly on theresults obtained by the French school of prob-abilities in the 1980’s (P. A. Meyer, J. Jacod) andby the statisticians of the Scandinavian school.

Statistics help determine the chromo-some carrying the gene

Once the existence of a gene for suscep-tibility to breast cancer is established by theanalysis of pedigrees, the second stage con-sists in locating it, at least roughly, on one ofthe 23 human chromosomes. For this the tech-nique of Œmarkers‚ has been available sincethe 1980’s; these are small, well-defined chainsof DNA which one can ``read” without muchtrouble, say by a fast chemical analysis. TheseŒmarkers‚ kinds of beacons, which are rela-tively easy to locate, make it possible, forexample, to determine any resemblancebetween areas of chromosomes under exam-

Finding a cancer-causing gene 29

Figure 1. A family in which one observes a concentration of breast cancer. The squares indi-cate men, circles women. Individuals are shown in black if they are affected, crossed out ifthey are deceased. Note that the grandmother, one of her daughters and three of her grand-daughters had cancer. Of course, the disease can still show itself among other members ofthe family. It is based upon such pedigrees that the geneticists were led to believe that thereis a gene for susceptibility to the disease.

ination in patients and their relatives. Thegreater the similarity in the same area of chro-mosomes of related patients, the higher is theprobability that this area carries a geneinvolved in the disease.

But such a statistical analysis is complicatedby the fact that parents do not transmit totheir children the chromosomes which theyhad themselves inherited from their parents,but rather a recombination of them (figure 2).If we consider two genes, which are initiallylocated on the same chromosome, they maywell be found on two different chromosomesafter recombination; the further the genes inquestion are located from each other, thehigher is the probability of this happening.Analysing the rate of similarity along a chro-mosome is consequently studying a randomprocess. Thanks to the statistics of processes,one can thus localise an interval in which agene of susceptibility is located. The use ofmarkers allowed the American team of JeffM. Hall, in Berkeley, to locate the gene BRCA1on chromosome 17 in 1990.

Reading DNA molecules to completelydescribe the gene and its abnormalforms

It is then a question of locating the genewith precision and of determining its struc-ture. It is known that DNA, the genetic mate-rial, is a long molecular chain ``written” in analphabet of 4 ``letters” (A, C, G and T, the ini-tials of the four types of molecules which con-stitute the DNA chain). Genetic data banksindex several billion of such sequences (some25 million letters are added every day...).

The precision of the method of markers, atbest, allows localisation of a gene in a DNAsequence containing some 4 million letters. Tofind out exactly which allele, or which mutationis responsible, for example, for breast cancer, it isnecessary to ``read” these sequences in healthysubjects and in patients and to compare them.That amounts to finding a ̀ `typographical error”in a text of 4 million characters, say a book of 2000pages, or rather in as many books of 2000 pagesas there are individuals to study. This is a difficulttask, even with powerful data-processing tools.It so happens that in man, genes constitute no

more than 3% of the chromosomes. Theremainder of the chromosomal materialis called intergenic. If one manages tolimit the search for typographical errorsto genes alone, one can reduce thesequence to be explored to about thirtypages, which becomes accessible by anycomputer.

But how to distinguish genes fromthe rest? It so happens that the ̀ `style”in which genes are written differs fromthe intergenic style: the frequenciesof successions of letters are not thesame. One can seek to exploit this dif-


Figure 2. In each pair of chromosomes of an individual, one chromosome is inher-ited from the father (in black) and the other is inherited from the mother (in white).A parent transmits only one chromosome from each pair to each offspring. Butbefore being transmitted, the chromosomes of each pair can exchange some of theirparts at random. This process, known as recombination, means that a parent trans-mits a recombined chromosome to the offspring (one of the four possibilities shownin the figure in which we assume that the chromosomes exchange two areas).

ference in style to annotate the sequence andto distinguish genes from the intergenic part.It is a difficult challenge. One must make useof statistical models called hidden Markovchains developed in the 1980’s, in connectionin particular with problems of automaticspeech recognition; they had to be adaptedto genomics, at the same time as one devel-oped algorithms capable both of characteris-ing the various styles and of assigning a styleto each position on the chromosome.

It is in this way that the precise locationof BRCA1 was found. Nowadays, one can eas-ily read it in each patient. This gene for sus-ceptibility to breast cancer has 5 592 lettersand one knows more than 80 of its alleles.But there remains a new challenge for thestatistician: to establish the relationshipbetween the various alleles and the preva-lence of this cancer.

Biology offers a new field of applica-tion for mathematics

As the example of the gene BRCA1 sug-gests, biology will probably play with respectto mathematics the role played by physics dur-ing a good part of the 20th century: to offera field of application for recent theoreticaltools and to spur the creation of new tools(we mentioned statistical tools here, but wecould have mentioned other fields of mathe-matics like dynamical systems, optimisation,even geometry - one knows that the spatialconfiguration of molecules plays an essentialrole in their function). Today, the statisticianfaces a new challenge: one is currently capa-ble of placing thousands of reagents on a glasssurface (``chip”) of one centimetre square andof thus knowing which genes work on which

tissues, under which experimental conditionsor... in which cancerous cells. Measurementstaken in the laboratory, under hundreds ofdifferent conditions, provide the researcherwith a considerable amount of numerical data,which characterise the expression of thou-sands of genes. To date, only statistical analy-sis can claim to deal with it and thereby deter-mine the relationship between genes anddiseases.

Bernard PrumLaboratoire Statistique et Génome

(UMR CNRS 8071),La Génopole, Université d’Évry

Finding a cancer-causing gene 31

Some references:

• B. Prum, « Statistique et génétique » dansDevelopment of Mathematics 1950-2000(sous la dir. de J.-P. Pier, Birkhäuser, 2000).

• C. Bonaïti-Pellié, F. Doyon et M. G. Lé, « Où en est l’épidémiologie du cancer en l’an2001 », Médecine-Science, 17, pp. 586-595 (2001).

• F. Muri-Majoube et B. Prum, « Une approchestatistique de l’analyse des génomes », Gazettedes mathématiciens, n° 89, pp. 63-98(juillet 2001).

• B. Prum, « La recherche automatique des gènes »,La Recherche, n° 346, pp. 84-87 (2001).

• M. S. Waterman, Introduction to computationalbiology (Chapman & Hall, 1995).

Whether they are stored digitally in computer memories,or they travel over the Internet, images take up a lot of space.

Fortunately, it is possible to ``condense''them without changing the quality!

a digital image can be compressed, just asorange juice can be reduced to a few gramsof concentrated powder. This is not by sleightof hand, but by mathematical and computerscience techniques making it possible to reducethe amount of space occupied by an image incomputer memory, or in communicationcables. Nowadays, these techniques are essen-tial for storing information, or for transmit-ting it by Internet, telephone, satellite or anyother means.

The compression of an image amounts torepresenting it using a smaller number ofparameters, by eliminating redundancies. Aexaggerated example will help in under-standing the basic principle: in the case of auniformly white image, it is unnecessary toexplicitly specify for each one of its points thegrey level at that point; that would take muchmore space than to simply state: ̀ àll the pointsof the image are white''. The problem of rep-resentation is central in mathematics, and its

Waveletsfor compressing images

Figure 1. These three images illustrate the power of current compression methods. The original image (A) consists of 512 x 512 points, each ofwhich has a certain level of gray taken from a palette of 256 levels. Image (B) is the result of a compression by a factor 8, obtained by reducing thelevels of gray to 2 possible values only (black or white). Image (C) was obtained from (A) upon compressing by a factor 32 by using a wavelet basis.The difference in quality from the initial image is hardly perceptible. (Illustration by the author)

A B C

applications go well beyond data compres-sion. During the last ten years, considerableprogress has been made thanks to the devel-opment of the theory of wavelets. In the fieldof image processing, this progress has led tothe adoption of the new standard of com-pression JPEG-2000. This is a meandering tale,which well illustrates the role of mathemat-ics in the modern scientific, or technologicallandscape.

Thirty-two times less space thanks towavelets

Let us consider an image such as that ofFigure 1A. It consists of 512 x 512 points, whoselevels of grey can vary from 0 (black) to 255(white). Each of the 256 possible levels of greycan be represented by a byte, i.e., a binary num-ber made up of 8 bits (a byte is thus simply asuccession of 8 bits 0 or 1, for example11010001).

One thus needs 512 x 512 x 8 = 2097152bits to encode a single image of this kind -which is a lot! The first idea which comes tomind to reduce the number of bits is to decreasethe number of levels of grey, for example, bylimiting oneself to white and black, as in Figure1B. The two possible values of the level of greyare encoded with only one bit (either 0 or 1),and one has thus decreased the number of bitsby a factor of 8. Obviously, the quality of theimage has deteriorated quite a bit. Now lookat the image of Figure 1C.

It has been encoded with 32 times fewerbits than the original image, by a method usingthe theory of wavelets; the deterioration ishardly perceptible! Why? Because instead ofreducing the precision, it is the manner of rep-resenting the information which was changed.

It all started with the analysis ofJoseph Fourier...

As we have said, a digital image is definedby 512 x 512 numbers which specify the lightintensity at each point. One can thus think ofthis image as a point in a space of 512 x 512dimensions - in the same way that a point ona surface, a two dimensional space, can belocated by two co-ordinates - and to ask whichco-ordinate axes are best adapted for repre-senting such a point. A system of axes (of amore abstract nature here than the familiaraxes of elementary geometry) defines whatone calls a basis.

A first fundamental advance was made bythe mathematician-physicist Joseph Fourier in1802, in his report to the the Académie desSciences on the propagation of heat, a subjectwhich is a priori unrelated to our problem.Fourier showed, in particular, that to repre-sent in a compact and convenient way the func-tion f(x) (from a mathematical point of view,such a function is a point in a space havinginfinitely many dimensions) one can use ̀ àxes''made up of an infinite set of sinusoidal func-tions. More precisely: Fourier showed that onecan represent a function f(x) as a sum of infi-nitely many sine and cosine functions of theform sin(ax) or cos(ax), each one carrying a cer-tain coefficient.

These ``Fourier bases'' became an essen-tial tool, frequently used in science, becausethey can be used to represent many types offunctions, therefore many physical quantities.In particular, they are also used to representsounds and images. And yet, engineers knowwell that these sinusoids are far from beingideal for signals as complex as images: they donot efficiently represent transitory structuressuch as contours of the image.

Wavelets for compressing images 33

...then came the wavelet transform

Specialists in signal processing were notthe only ones to become aware of the limita-tions of Fourier bases. In the 1970’s, a Frenchengineer-geophysicist, Jean Morlet, realisedthat they were not the best mathematical toolfor underground exploration; this led to oneof the discoveries-in a laboratory of Elf-Aquitaine-of wavelet transform. This mathe-matical method, based on a set of basic func-tions different from the sinusoidal functionsused in Fourier's method, advantageouslyreplaces the Fourier transform in certain situ-ations. In addition, already in the 1930’s, physi-cists had realised that the Fourier bases werenot well-adapted for analysing the states ofan atom. This spurred much work, which lateron contributed much to the theory of wavelets.It was also in the 1930’s that mathematiciansstarted trying to improve the Fourier bases foranalysing localised singular structures, whichopened an important research program stillvery much alive today.

In other words, a multitude of scientificcommunities developed modifications ofFourier bases with the means at their disposal.

In the 1980’s, Yves Meyer, a French mathe-matician, discovered the first orthogonalwavelet bases (orthogonality is a propertywhich considerably simplifies reasoning andcalculations; Fourier bases are also orthogo-nal). This discovery, followed by some unex-pected meetings around photocopiers or cof-fee tables, started a vast multi-disciplinaryscientific movement in France, which has hada considerable impact internationally. Theapplications of the theory and algorithms ofwavelets have made their way not only intomany scientific and technological disciplines,but have also led to the creation of severalcompanies in the United States.

Mathematics of waveletshas played a pivotal rolein a number of fields

Mathematics has played a fundamentalrole here as a catalyst, and in the clarificationand deepening of ideas. By isolating the fun-damental concepts from specific applications,it allowed scientists from very diverse fieldsof physics, such as signal processing, computer

science, etc. to realise that theywere working with the same tool.Modern mathematical work onFourier analysis has now permittedus to go further and to refine thesetools, and to control their per-formance. Finally, this theory pro-vides a standard technique for sci-entific computation (the fastwavelet transform), thanks to a col-laboration between mathemati-cians and specialists in signal pro-cessing. The image of Figure 1C wasthus obtained thanks to the samewavelet bases as those used in sta-


Figure 2. The graph of a wavelet used in the compression of images.

tistics, seismology, or scientific computation,with the same fast algorithm. And throughthe international standard JPEG-2000 for thecompression of images, these wavelets havecurrently invaded all fields of imaging, fromthe Internet to digital cameras, and is mov-ing towards satellites.

A bridge remains to be built betweenthe world of wavelets and the worldof geometry

Fourier bases were not well-adapted tothe analysis of transitory phenomena, whereaswavelet bases are. Is this the end of the story?No. In image processing, as in all other fieldswhere wavelets have became a basic tool,everyone currently confronts the same typeof problem: how to exploit geometrical reg-ularities. Indeed, we know that an image,however complex, is remarkably well repre-sented by a simple drawing made up of rela-tively few strokes, and one can often think ofthe contours of the objects appearing in theimage as being made up of rather simple geo-metrical curves. Using profitably these curvesand their regularity should make it possibleto improve considerably the results obtainedup until now; but wavelet theory is not atpresent capable of this. To build this bridgewith the world of geometry poses difficultmathematical problems. However, the scien-tific and industrial stakes being high, one canexpect that it will be built in the coming tenyears. In France?

Stéphane MallatDépartement de mathématiques appliquées,

École polytechnique, Palaiseau

Wavelets for compressing images 35

Some references:

• B. B. Hubbard, The world according to wavelets(AK Peters, 1996).

• S. Mallat, A wavelet tour of signal processing(Academic Press, 1999).• Y. Meyer, Wavelets and Operators(Cambridge University Press, 1996).

Daniel Bouche

How to escape being detected by radar? What is the optimal shape of a sound-proof wall?

Can one improve the quality of sonographic images?To get a satisfactory answer, these questions require

a thorough theoretical analysis.

what is a wave? Anyone who can give aprecise and unique answer to this questionwould be quite clever! All the same, waves areeverywhere, and a great number of scientistsand engineers come across them on a dailybasis. In somewhat vague and intuitive terms,one can say that a wave is the propagation ofa signal, of a perturbation, in a certainmedium, at an identifiable speed.

There is no lack of examples. There are, ofcourse, the wavelets, which one can create onthe surface of water by throwing a small stoneinto it; here, it is a perturbation of the heightof water which is propagated. The distancebetween two successive wavelets is the wave-length, a fundamental quantity in the des-cription of wave phenomena. As far as soundwaves are concerned, they bring into playvariations of pressure and density of theambient medium (air, most often), these varia-tions occurring at audible frequencies.Acoustic waves are of a similar nature, andinclude both sound waves and those which

the ear cannot perceive. When waves propa-gate in a solid, one talks about elastic waves,such as the seismic waves which traverse theinterior of our planet and which are detectedby seismographs.

The case of electromagnetic waves is par-ticularly important. These are variations ofelectric and magnetic fields which propagatein the vacuum at the speed of light. Visiblelight, infrared rays, ultraviolet rays, X-rays,gamma rays, microwaves, radio waves, radarwaves, all these phenomena are examples ofelectromagnetic waves. What differentiatesthem is their frequency, or their wavelength(which varies from a fraction of a micrometrefor visible light, even less for ultraviolet rays,gamma rays and X-rays, to a few centimetres,or a few hundred meters for radar and radiowaves).

The study of wave behaviour not onlyhelps us to understand nature surroundingus, but also helps us to master a number of

Preventing wavesfrom making noise

technologies and, consequently, leads us tonew inventions. The behaviour of light wavesaffects the whole gamut of optical instru-ments, be they photographic lenses, micro-scopes, telemetry devices, etc. One can wellimagine knows that radar waves have mili-tary applications, in the design of stealthymilitary vehicles, i.e., those which escapedetection by radars as much as possible. Asfor acoustic waves, one can cite the designof concert halls having optimal acoustics, ofsound-absorbing materials or structures, ofactive anti-noise devices (i.e. those which emitsound waves opposite to those of the noisein order to neutralise it), of machines for sono-graphy or for destroying kidney stones, ofnondestructive testing devices (for detectingdefects in some part of an aeroplane, forexample), etc.

Known equations, but difficult toexactly solve accurately

The equations which govern the varioustypes of waves have been well-known for along time. Thus, those relating to electroma-gnetic waves were established by the Scottishphysicist James Clerk Maxwell more than acentury ago, around 1870. But it is not enoughto know the equations obeyed for exampleby a radar wave to find out how this wavepropagates, or interacts with an obstacle suchas an aeroplane, or any another object whichwe want to detect and locate, and gets par-tially reflected towards the radar antennawhich emitted it. For that, it would be indeednecessary to solve these equations in whichthe unknown is the undulatory field, i.e., theamplitudes of the wave at each point of spaceand at every moment. That is not at all easy.We are dealing here with partial differentialequations (in which appear the unknownamplitude of the wave and its derivatives with

respect to the space and time coordi-nates), which have to be supplemen-ted by ``boundary conditions''. Theseconditions give a mathematical des-cription of essential data such as thevalue of the undulatory field at the ini-tial moment, the shape of the obstacleand the way in which the wave behaveson its surface (reflection, absorption,etc), the way in which the amplitudeof the wave decreases at very great dis-tances from the source and the obs-tacle.

The resolution of these kind of pro-blems where the wave is diffracted(deviated or modified) by objects iscomplex; it requires mathematicaltools, some of which are simple and

Preventing waves from making noise 37

The Petit duc is a drone (small radio-controlled aircraft) which is under deve-lopment at Dassault Aviation. It is a furtive aircraft: its form and materials havebeen chosen so that it is difficult for it to be detected by radar waves. The choicehas been made on the basis of complicated calculations based on wave propaga-tion; in certain cases, the accuracy of such calculations leaves something to bedesired and is the subject of continued research (Photo Dassault Aviation).

have been known for a long time, others aremuch more sophisticated and are still beingdeveloped. In fact, in a more general context,partial differential equations constitute a veryimportant branch of mathematics, which hasbeen the subject of active research for morethan two hundred years. Once the equationsand their boundary conditions have been esta-blished, one of the first tasks of the mathe-matician consists in formulating the problemin rigorous terms and showing that the equa-tions have a solution, and that if such is thecase, the solution is unique (otherwise, it wouldmean that the problem is badly defined, thatthe model is incomplete). Such a study can bedifficult, and one cannot always carry itthrough; but it allows us to ensure that cal-culations leading to a solution will not be invain!

Mathematical analysis makes a rigo-rous formulation of the problem pos-sible and allows the development ofeffective methods for its solution

The question narrows boils down to pro-posing effective and accurate methods to solvethe resulting problem with sufficient preci-sion. Except for very simple cases, a so-calledanalytical solution, where one obtains an exactand general result expressed by a compact for-mula, it is out of reach. Scientists and engi-neers have to be satisfied with a numericalsolution, carried out on a computer becausethe necessary calculations are unwieldy, whichgives the result in the form of numerical values(numbers), valid with a certain approximation.Significant difficulties appear here too.

Thus, in problems bringing into play dif-fraction of waves by objects, the propagation

medium is often unlimited: the wave can goall the way to infinity. However, in order toretain the uniqueness of the solution, it isnecessary to impose a condition known as theradiation condition, which specifies how theamplitude of the wave decays as it moves away.This condition is not simple to implementnumerically. One of the proposed solutionsconsists in transforming the original partialdifferential equation into an integral equa-tion (an equation in which the unknown func-tions appear in integrals); the advantage ofthis formulation is that it automatically satis-fies the radiation condition.

It was in the 1960’s that the first compu-ter programmes for the numerical solution ofintegral equations were written. They couldonly calculate diffraction by objects which


A typical problem of wave propagation: a source S emits a radar,light, acoustic or some other kind of wave (in red in the figure) of adefinite wavelength; the wave is reflected partially (in blue and greenin the figure) by the two obstacles at O1 and O2; what will be theamplitude of the resulting wave in each case, for example, at the detec-tor placed at S? The solution of this difficult problem must take intoaccount the type of wave emitted, its wavelength, the shape of the obs-tacles, the material of which they are made, etc.

were small compared to the wavelength;moreover, as sufficient mathematical analysiswas lacking, they often gave aberrant results.Starting from the end of the 1980’s, a betterunderstanding of the problems encounteredand their solution allowed the calculation ofthe diffraction of a wave by larger and largerobjects compared to the wavelength, withgreater and greater precision. Research iscontinuing in various fields today, such as thechoice of the integral formulation best adap-ted to the problem and better numerical tech-niques to solve the equation. In particular,methods known as multipolar methods havemade it possible to considerably increase thesize of the problems which can be handled.This work has contributed to the developmentof reliable software tools capable of calcula-ting with precision the undulatory field dif-fracted by objects whose size may go up toseveral dozen times the wavelength. Such is,in particular, the case of an aeroplane in thefield of a decimeter-wavelength radar.

A method rival to the formulation in termsof integral equations consists in directly sol-ving the partial differentialequations and to free oneselffrom the radiation conditionby artificially limiting the pro-pagation medium by an`àbsorbing boundary condi-tion'': one posits (mathemati-cally) the presence of an ima-ginary border, whichcompletely absorbs all wavesthat affect it. For a long time,these absorbing boundaryconditions were responsiblefor the presence of parasiticreflection phenomena in thenumerical solution; they were

particularly awkward in the case of weakly dif-fracting objects. But these numerical tech-niques, making use of absorbing boundaryconditions, have also made considerable pro-gress; they now permit a very weak level ofparasitic reflection, thanks to theoretical workdone mainly at the beginning of the 1990’s.

Geometrical optics and itsgeneralisations at the serviceof short wavelengths

When the size of the obstacle diffractingthe wave is very large compared to the wave-length (such as a droplet of water illumina-ted by visible light, or a plane swept by a deci-metre-wavelength radar, etc.), there is aneasier way than the direct solution of the waveequation: good old geometrical optics. Oneconsiders light waves as rays propagating ina straight line in a given medium and whichare subject to the simple laws of reflectionand refraction, which were discovered seve-ral centuries before the equations describingelectromagnetic waves. One of the contribu-

Preventing waves from making noise 39

Waves propagating on the surface of water: even this daily and common phenomenon can beextremely difficult to describe correctly and precisely. (Photograph: Getty Images)

tions of physicists, in particular of the Germanphysicist Arnold Sommerfeld (1868-1951), wasto show that geometrical optics is ultimatelya way of solving diffraction problems whenthe objects are infinitely large compared tothe wavelength.

But of course the size of real objects isnever infinite: geometrical optics is thus justan approximation which may at times be good,at times not so good. Therefore, it has beenextended and generalised in order to deter-mine the undulatory field at points where tra-ditional geometrical optics would considerbeing in the shade. This work, which startedin the 1950’s, continues even today; it makesit possible to have tools which are certainlyless precise than the method of direct nume-rical solution of partial differential equationsbut which are applicable in the short-wave-length regime.

Despite all these advances, many problemsof wave motion have still not been solved satis-factorily. Such is the case of diffraction byobjects whose size is large compared to thewavelength but which have a complex form,with fine features comparable in size to thewavelength (for example a plane or a missilewhen one wants to take into account theirdetailed form down to the last bolt, and notjust their general shape). Hybrid methodsshould be able to do the work, but there stillremains much to do in this field and elsew-here!

Daniel BoucheCEA (Commissariat à l’énergie atomique),

Département de physique théorique et appliquée,Direction d’Île-de-France


Some references:

• INRIA wawe project“Ondes W3 server’’ :http://www-rocq.inria.fr/poems/index.php?langue=en

• G. B. Whitham, Linear and non-linear waves(Wiley, 1974).

• D. S. Jones, Acoustic and electromagnetic waves(Oxford University Press, 1986).

• J. A. Kong, Electromagnetic wave theory(Wiley, 1990).

• E. Darve, « The fast multipole method: numeri-cal implementation », Journal of ComputationalPhysics, 160 (1), pp. 195-240 (2000).

• I. Andronov, D. Bouche, “Asymptotic and hybridmethod in electromagnetics” (IEE, 2005).

Francine Delmer

Scientists are not the only ones to be inspired by mathematics.Many artists have drawn the subject of some of their works from it.

The converse is also sometimes true, as in the case of perspective,where art has led to geometrical theories.

From November 2000 to January 2001,the Galerie nationale du Jeu de Paume pre-sented a retrospective of the artist FrançoisMorellet, who was called a ``postmodernPythagorician'' by the the criticThomas McEvilley in the cata-logue of the exhibition. InFebruary 2001, Tom Johnson isawarded the Victoire de lamusique for contemporary musi-cal creation for his work KientzyLoops. This composer works outmusical transpositions fromsuites which act like constraints,diverts automatons, plays outPascal's triangle (Self-ReplicatingLoops, Rhythmic canons, etc.).He always places mathematicalconcepts at the forefront of hiswork, and has been having a longdialogue and fruitful exchangewith Jean-Paul Allouche, resear-cher in number theory and theo-retical computer science. In the

same year, David Auburn's Proof, which bringsthe lives of mathematicians onto the stage,wins the Pulitzer prize for drama. Written forthe layman, this work offers an interesting

When art rhymeswith math

It is said that Galileo once, instead of listening to a sermon in the cathedral in Pisa, obser-ved the swinging of the chandelier hanging from the ceiling. He had the idea of countingthe oscillations and noticed that their frequencies were different, but that they were inver-sely proportional to the square root of the length of the pendulum. It is on this observationthat the work Galileo of the composer Tom Johnson is based. Here, the pendulums are sus-pended from a structure designed and built by the artist-engineer Eric Castagnès of Bordeaux.(Photo Eric Castagnès)

vision of the work of a researcher and putsforward certain characteristics of this milieu.One can detect in it allusions to the recent andsingular history of the American mathemati-cian John Forbes Nash and to the proof ofFermat's last theorem by the British mathe-matician Andrew Wiles.

These three events, reported by the media,illustrate the topicality of the mutual fascina-tion between mathematicians and artists.Throughout history, their relations have tra-versed the whole gamut of artistic fields andtheir discussions have been carried out on avariety of levels, as testified by philosophers,art historians, epistemologists, artists andmathematicians when they discuss the realityand relevance of mathematics. Our purposehere is not of legitimizing any artistic creationbecause of its references to scientific theories,nor of making value judgments, nor of attemp-ting any classification of mathematical or artis-tic practice. We will only confine ourselves tothrowing some light on the mutual relation-ship in the manner of a pointillist.

The links between mathematicsand the arts woven over time

It is said that approximately 2700 yearsbefore our era, the Egyptians were using thesacred right-angled triangle with sides 3, 4, 5(these numbers satisfy the relation ̀ `square ofthe hypotenuse = sum of the squares on theother two sides'' which characterises right-angled triangles) in the construction of pyra-mids. One can also think of the Pythagoriantheories, of around 500 B. C., according towhich musical harmony is governed by certainratios of numbers. Closer to our time, AlbrechtDürer and Leonardo da Vinci, emblematic

figures of the humanistic spirit of theRenaissance, were interested in geometry,optics, architecture and the theoretical, as wellas, practical questions inherent in these fields.Dürer, inspired by the thinking and work ofthe Italians, in particular Piero della Francescaand Alberti, fixed the rules of perspective inhis geometrical treatise Underweysung dermessung (1525). Since then, artists have syste-matically used it in theirs works, while theFrench mathematicians Girard Desargues andGaspard Monge developed projective and des-criptive geometries in the XVIIth and XVIIIth cen-turies. It should be noted in this particular casethe precedence of art over science: as the arthistorian Eric Valette puts it, ̀ `the invention ofperspective is certainly one of the most obviousexamples where the symbolic artistic systembrings knowledge of the world still unknownto science''.

In literature, mathematics might appearto be less present. However, the members ofOulipo (Ouvroir de littérature potentielle, orWorkshop of potential literature, founded in1960 by Raymond Queneau and François LeLionnais, writers and mathematicians), oftendraw their themes from it. Thus, in Life: User'smanual by George Perec, the plot rests uponthe solution of a combinatorial problem invol-ving an order-ten orthogonal bi-latin square.

Musical creation in the 20th century wasmarked by the composers Pierre Boulez andIannis Xenakis, both of whom were trained inmathematics. In his compositions Boulez deve-lops the principles of serialism, while Xenakiscalls upon a statistical control of the musicalparameters in his stochastic music, to quoteonly one example of their creations. The IRCAM,created in 1970 by Pierre Boulez, where a largenumber of musicians, acousticians, mathema-


ticians and computer scientists with a mixedtraining work, is testimony to the major inter-play between mathematics and music at thebeginning of the 16th century, both at the tech-nical, as well as, the theoretical level. Questionsrelating to this subject were posed from aninteresting perspective at the FourthMathematical Forum Diderot held there inDecember 1999 by the European MathematicalSociety under the title Mathematical logic, musi-cal logic in the 20th century.

Mathematics, a simple tool as well asa theoretical drive of creation

These examples illustrate the diversity ofthe relations between mathematics and thearts, and raise some questions. Is mathema-tics used, by such and such art, for technicalor theoretical reasons? Does it inspire artists

in a metaphorical, or a symbolic way?

The painter François Morellet, who hasalready been mentioned, uses mathematicsmore as a tool, as shown by his Répartitionaléatoire de quarante mille carrés suivant leschiffres pairs et impairs d'un annuaire de télé-phone [Random distribution of forty thousandsquares according to the even and odd digitsin a telephone directory], πι ironicon n° 2, etc.,where he suggests the idea of infinity.

According to the art critic GillesGheerbrandt, ̀ `with him, (elementary) mathe-matics may be used for the formulation of pro-blems, but it is a simple tool, never an end initself''. As for the artist, he claims using mathe-matics to escape any subjectivity or emotion,to keep a distance from his work, to desensi-tise it; in this he revisits the old Platonic ideo-

When art rhymes with math 43

The artist François Morellet, a ``postmodern Pythagorician''. (Photo Gamma/Raphaël Gaillarde)


logy, which claimed art was nothing but decep-tion.

If certain artists use elementary conceptsas a reference or pretext, others adapt prin-ciples of mathematical theories as their basis,thus drawing upon the essence of reasoning.The painter Albert Aymé, one of the most radi-cal examples of an excursion into abstraction,takes a path similar to that of research inmathematics.

Rejecting combinatorial mechanisms, hedevelops his thoughts in treatises - Approachof a specific language, On paradigms, etc. -which provide the framework for his pictorialproject: `Ì try to continue working with therigour of a scientist but without dissociatingmyself from the passion of the poet or themusician''. Moreover, the work does not needwords and remains `ìntrinsically beautiful'',abstract art being ``not a matter of taste butof method'', according to him.

As human activities, mathematics and thearts are products of individuals surrounded bythe same cultural, political, religious climate.The great upheavals of history do not leaveany of these fields untouched because of inter-actions which seem dependent on the spiritof the time.

Was it not upon reading the philosophi-cal writings of Henri Poincaré, who populari-sed at the turn of the 20th century the ideasof non-Euclidean geometry, that the cubistsdid away with traditional perspective?

Let us keep in mind that any attempt atthe fusion or unification of mathematics andthe arts would be reductionist and in vain. Itis indeed knowledge and curiosity which per-mit exchanges and confrontations between

them, in a way specific to each form of expres-sion.

Let us just make the happy observationthat mathematics and the arts are still playingtogether, a piece of music called enlighten-ment.

Francine DelmerLaboratoire Arithmétique et Algorithmique

expérimentaleUniversité Bordeaux 1, Talence

Some references:

• E. Valette, La perspective à l’ordre du jour(L’Harmattan, 2000).

• G. Gheerbrant, « François Morellet », Parachute,Montréal, n° 10, p. 5 (printemps 1978).

• M. Loi (sous la dir. de), Mathématiques et arts(Hermann, 1995).

• J.-L. Binet, J. Bernard, M. Bessis (sous la dir.de), La création vagabonde (Hermann, collec-tion Savoir, 1980).

• V. Hugo, L’art et la science (Anaïs et Actes Sud,1864/1995).

• M. Sicard (sous la dir. de), Chercheurs ou artistes(Autrement, série Mutations, n° 158, 1995).

• I. Xenakis, Arts/sciences. Alliages (Casterman, 1979).

• J.-M. Lévy-Leblond, La pierre de touche - lascience à l'épreuve (Gallimard, 1996).

• J. Mandelbrot, « Les cheveux de la réalité - auto-portraits de l’art et de la science », Alliage, 1991.

• D. Boeno, « De l’usage des sections coniques »,Cahiers art et science, n° 5, pp. 41-54(Confluences, 1998).

45


Nguyen Cam Chi and Hoang Ngoc Minh

The biological activity of a DNA molecule depends mainlyon the way it is arranged in space and the way

in which it is twisted - things which fall within the provinceof the mathematical theory of knots.

everyone now knowsthat DNA is the mole-cule which carries thegenetic information ineach cell of living orga-nisms and is responsiblefor cellular activity to alarge extent. In general,DNA consists of twolong parallel strandsmade up of a sequenceof molecules callednucleotide bases, thetwo strands turningaround one another toform a helicoidal struc-ture: the famousdouble helix.

The information carried by DNA is enco-ded by a sequence of pairs of nucleotide bases.This sequence does not depend on the way inwhich the molecule is twisted, tangled or knot-ted. However, after the discovery in the years

-1970 of circular DNA molecules (i.e., loopsmade up of only one strand or two strandswound around one another), one started won-dering about the influence of the topologicalshape of DNA, of its form in space. In 1971,the American biochemist James Wang provi-

From DNA to knot theory

A circular knotted DNA molecule, seen through an electron microscope. The topology of DNA mole-cules influences their activity. (Photo N. Cozzarelli, University of Berkeley)

ded evidence showing that certain enzymes,the topoisomerases, can modify the topolo-gical configuration of DNA, for example, bymaking it knotted, and that the topology ofthe DNA molecule influences its activity in thecell. The study of the topological configura-tions of DNA can thus tell us something aboutthe way in which DNA participates in cellularmechanisms.

Topology, defined by some as ̀ `rubber geo-metry'' - that is, the study of properties whichare not modified by deformations or by chan-ging lengths - is an important and fundamentalbranch of mathematics. Almost every mathe-matician needs its concepts and methods. Thetheory of knots is a particular branch of topo-logy. Born roughly a century ago, its aim is tomake a precise study of knots and to classifythem. It has found applications in other scien-tific disciplines (in molecular chemistry, statis-tical physics, theoretical particle physics, etc.),in addition to its links with other fields ofmathematical research.

The fundamental question of the theoryof knots is as follows: given two knots (whichare not too simple!), made up of thread, forexample, can one decide if they are equiva-lent? In other words, can one stretch or deformone of them to make it identical to the other,without cutting them up? As topologists allowdeformations, their definition of a knot isslightly different from that of the man in thestreet: for them, a knot is something obtai-ned by joining the two ends of the thread;otherwise one could - by pulling and chan-ging the form of the thread correctly - untieany knot, and all knots would be then equi-valent. From the point of view of topology,therefore, a knot necessarily consists of oneor more loops, which is the case for circularDNA.

To classify knots by finding `ìnva-riants'': a problem of algebraic topo-logy

Specialists in knot theory do, in general,algebraic topology: they try to associate toeach topologically different knot an `ìnva-riant'', a mathematical object which charac-terises it, which is easily computable and whichlends itself to algebraic manipulations. Thismathematical object can be, a priori, a num-ber, a polynomial (an algebraical expressionsuch as x6 - 3x2 + x + 2) or something more com-plicated and abstract. The important thing isthat it should be the same one for all topolo-gically equivalent knots (hence the term ofinvariant). The ideal would be to find inva-riants which characterise knots completely,i.e., such that two distinct knots have neces-sarily different invariants. Then the problemof classification will be solved. To summarise,the main questions are: is there a way of cha-racterising knots in order to distinguish them?Does there exist an algorithm to distinguishtwo knots? Does there exist a computer pro-gram which distinguishes two given knots ina reasonable amount of time?

In spite of several decades of research, theanswer to these questions remains incomplete.However important progress has been made.Let us mention some of it briefly. In 1928, theAmerican mathematician James Alexanderintroduced the first polynomial invariant (theAlexander polynomial) permitting the classi-fication of knots. But the Alexander polyno-mial is an incomplete invariant: certain dis-tinct knots have the same Alexanderpolynomial. Much more recently, in 1984, theNew Zealand mathematician Vaughan Jonesdiscovered a new invariant, which was also apolynomial; it is more effective than the


Alexander polynomial, but it doesn’t entirelysolve the problem of classification either. Sometime later, other researchers refined and gene-ralised the Jones invariant; still, the newlyintroduced polynomial invariants are incom-plete and fail to differentiate between cer-tain topologically distinct knots.

Perhaps the beginning of a complete solu-tion was made around 1990, with the workof the Moscovite researcher Victor Vassiliev.He introduced a new class of invariants defi-ned in an implicit way, i.e., only defined bythe relations which they satisfy amongst them-selves. The Vassiliev invariants are numbers,i.e., to each knot is associated a number (whichcan be determined based on a combinatorialanalysis of the topology of the knot). Vassilievconjectured that these invariants form a com-plete system, in other words, that distinct knotsalways have different Vassiliev invariants.Although no counterexample has been founduntil now, the conjecture remains to be pro-ved, just as effective methods to computethese Vassiliev invariants in an efficient wayremain to be found. All the same, progresshas been considerable.

There is a parallel betweenmathematical transformationsand enzymatic mechanisms

This mathematical research is related tothe questions posed by biologists in connec-tion with molecules such as DNA. For example,around 1973, the British mathematician JohnConway introduced elementary ``surgical''operations (the flip and the uncrossing), whichmake it possible to transform a knot into ano-ther one by modifying it at the level whereits strands cross each other. However, theseoperations of a mathematical nature have bio-chemical equivalents which are carried out bytopoisomerases. These enzymes, which areessential to the functioning of every cell, caninitially cut one of the strands or the twostrands of the circular DNA ring and pass asegment of the ring through the opening, andthen close up the ends which had been cut,to make a knot in each ring. By carrying outthe operations of cutting, passing throughand resoldering, they can cut a strand, makethe other strand pass through the opening soobtained, and then resolder the cut (this cor-responds to the Conway's flip operation), or

to make two cuts and tworesolderings by attachingthe two strands ``thewrong way'' (Conway'suncrossing operation).

Now, how can thetopology of DNAinfluence its biologicalactivity? Let us illustratethis with the example ofa supercoiled DNA mole-cule. In its usual state, thestrands of the moleculardouble helix describe a

From DNA to knot theory 49

The two knots represented here are topologically distinct: one cannot pass from one to the other by merelypulling the threads, without crossing and resticking. The Alexander polynomial of the one on the left (atrefoil knot) is P(t) = t2-t + 1; the Alexander polynomial of the one on the right is P(t) = t2-3t + 1. Thesetwo polynomials are distinct, as they should be. However, there are distinct knots which have the sameAlexander polynomial: the Alexander polynomial is not a complete invariant.

certain number of revolutions around the axisof the helix. Certain topoisomerases canincrease or reduce this twisting, just as onecan twist the telephone wire a bit more or abit less, which modifies its form. What's more,in circular DNA the number of revolutions ofthe double helix is a topologically invariantproperty: it cannot be changed by any defor-mation of the shape, except when the cuttingand reattaching of the strands of the two-coi-led DNA ring are involved. Now, if a DNA ringis unrolled, one can easily see that the doublehelix becomes less compact, and that its inter-nal parts become more exposed to the actionof the enzymes which surround it. Such expo-sure accompanies DNA replication (making asecond copy of the molecule) and its trans-cription (process by which proteins are syn-thesized in the cell).

As the topological configuration of DNAis determined by an enzymatic mechanism, alegitimate question for biologists would be toknow the extent to which the topological clas-sification of knots makes it possible to unders-tand the enzymatic mechanisms at work.Another related question would be to knowif one can simulate all these enzymatic mecha-nisms using the basic operations introducedfor mathematical knots. Research at the bor-der between the mathematics of knots andmolecular biology is now very active.

Nguyen Cam Chi et Hoang Ngoc MinhDépartement de mathématiques et

d’informatique,Université de Lille 2


Some references:

• « La science des nœuds », dossier hors-série dePour la Science, avril 1997.

• A. Sossinsky, Nœuds - Genèse d’une théoriemathématique (Seuil, 1999).

• D. W. Sumners, « Lifting the curtain : usingtopology to prob the hidden action ofenzymes », Notices of the AmericanMathematical Society, 42 (5), pp. 528-537(mai 1995).

Pierre Cassou-Noguès

Throughout their history, mathematics and philosophy have had a closeand enigmatic relationship. It would be necessary to go as far back asPlato in ancient Greece and Descartes at the dawn of modern times.

Let us cite here two great figures of the 20th century,David Hilbert and Edmund Husserl.

edmund Husserl and David Hilbert met inGöttingen in 1901. The philosopher Husserlhad studied mathematics. In Berlin, he wasthe assistant of Karl Weierstrass, a great mathe-matical analyst, before meeting FranzBrentano in Vienna and turning towards phi-losophy. In 1891 he published The Philosophyof Arithmetic. The first volumes of his LogicalResearches appeared at the same time as thephilosopher settled in Göttingen. The mathe-matician Hilbert was in Göttingen since 1897.He had solved a famous problem, ``Gordan'sproblem'' in the theory invariants, which hadpreoccupied German geometricians for thelast twenty years. He had developed the``theory of algebraic fields'' in algebra. Husserland Hilbert were roughly the same age, andthey ran into each other in the PhilosophyFaculty, which, in reality, regrouped philoso-phers and mathematicians. Each of them isgoing to transform his discipline. Husserl dis-covers phenomenology. Hilbert introduces theabstract method which characterises modernmathematics.

The philosopherand the mathematician

David Hilbert (1862-1943) was, along with the Frenchman HenriPoincaré, one of the great mathematicians during the 1900’s. By thedepth of his work and his points of view, by his dynamism that ins-pired Göttingen, he exerted a considerable influence on the mathe-matics of the 20th century. (Photo AKG)

Göttingen, centre of mathematicalexcellence, welcomes philosophers

Even though Göttingen was only a smalltown close to Stuttgart, shortly after 1900 itbecame the centre of the mathematical worldwith Felix Klein as the head of the Faculty. Thisgreat geometrician, who had definitely esta-blished the existence of non-Euclidean spaces,had given up active research and devoted him-self to teaching, to the development of mathe-matics in the 19th century, and to the admi-nistration of the Faculty, for which he foundnew funding. He recruited Hilbert and thenHermann Minkowski. This latter will introduce,on the occasion of a famous lecture, the``space-time continuum'' - which carries hisname and which will be used by Einstein asthe framework for formulating his theory ofrelativity. Every week, the Göttingen

Mathematical Society meets for a lecture, bysomeone from Göttingen, or from somewhereelse. Husserl, the philosopher, speaks in April1901 about the problem of the imaginaries inarithmetic. Göttingen is a place devoted tomathematics; It seems that one day Minkowski,walking in the main street, meets a pensiveyoung man tormented by some thought; hetaps him gently on the shoulder and says:``don't worry, young man; it converges'', atwhich point the young man moves away, reas-sured.

It was at Göttingen that Hilbert gave amature form to his abstract method of modernmathematics. The abstract method was bornin the algebra of the 19th century. RichardDedekind and Leopold Kronecker, in particu-lar, had introduced what are called structures.One defines a mathematical structure, such as


One of the mathematics buildings (the Institute for applied and numerical mathematics) of the university of Göttingen, today. Between 1900and 1930, Göttingen was a world-famous centre for mathematics, thanks to the efforts of David Hilbert. Mathematicians there would oftenmeet philosophers and scientists from other disciplines. (Photo University of Göttingen)

that of a ``group'', a ``vector space'', or a``field'', by specifying the rules which the ope-rations verify, without going into the natureof the objects on which they operate. Thus,the same structure can apply to objects of dif-ferent nature - to numbers, to functions, orto geometrical transformations, etc.Abstraction in mathematics consists of igno-ring, or making an abstraction of, the natureof objects and considers only the relations bet-ween these objects. This point of view, whichhad emerged in the algebra of Dedekind andwhich remained unstated in the 19th century,is made explicit by Hilbert.

Hilbert gives an axiomatic presenta-tion of geometry

As soon as he arrived in Göttingen, Hilbertannounced a course in geometry. This course,which will be published under the title Thefoundations of geometry, relies upon the abs-tract method to give an axiomatisation of geo-metry. Hilbert disregards the nature of geo-metrical objects, points, lines, planes, and iscontent with specifying the relations betweenthem, relations whose properties are given bythe axioms. In other words, axioms fix the pro-perties of the relations existing betweenobjects whose nature is left unspecified. Thus,the axioms define a structure similar to thealgebraic structures. But, in going from alge-bra to geometry, the primacy of structures isreinforced. In algebra, one gives a structureto supposedly known objects, such as num-bers and functions. One can deduce a theo-rem by reasoning about the structure, or byreasoning about the objects which have theirown specific nature. On the other hand, byaxiomatisation, reasoning is reduced to asimple deduction starting from the axioms,

and the objects are defined by specifying theaxioms. Axioms, i.e., the structure, suffice todefine the objects and to carry out reasoningon these objects.

In his axiomatisation of geometry and inhis later work, Hilbert makes the abstractmethod of algebra explic, makes it radical anduses it to produce new results. Actually, Hilberttraverses and transforms the whole of themathematics of his time with this abstractperspective: geometry; algebra and the theoryof numbers, with a first proof of ``Waring'sconjecture'' in 1909; analysis, where he intro-duced Hilbert spaces, abstract spaces whose``points'' are, for example, functions. The abs-tract method will be taken up in Göttingenby the school of Emmy Nœther and Emil Artin,and then in France by the Bourbaki group. Itwill nourish from then on the whole of mathe-matics.

Providing a foundationfor mathematics

In parallel, Hilbert was developing the abs-tract method to launch a program of foun-dations for mathematics. To provide a foun-dation for mathematics is to give an ultimateguarantee to mathematical reasoning. Itmeans, in particular, to justify arguments whichpresuppose the actual existence of an infinity,among them transfinite induction, while atthe same time avoiding the assumption of theexistence of infinity. The formalist programmeconsists of two steps. The first task is to for-malise mathematical theories. We consider analphabet of symbols. We fix rules analogousto those of spelling and grammar, to constructformulas with these symbols. We specifyaxioms which will serve as premises in the

The philosopher and the mathematician 53

proofs, and rules for deducing a formula fromothers. Mathematics is replaced by a set of for-mulas. Proof consists in manipulating thesesymbols according to explicit rules, disregar-ding the meaning of the symbols. A proof islike an assembly of symbols which conformsto explicit rules, a drawing made according togiven rules. The second task is to show thenon-contradiction of these formal systems bymeans of finite reasoning, i.e., not utilisingthe actual infinity.

The first theory to which Hilbert tried toapply this programme was arithmetic, whichalready contained transfinite reasoning. Thus,Hilbert inaugurated proof theory, which consis-ted of finite reasoning about drawings whichrepresented proofs in a formal system.However, in 1931, the Austrian logician KurtGödel established that it is impossible to prove,by means of finite reasoning, the non-contra-diction of a formal system which includes ele-mentary arithmetic. It is thus necessary to giveup Hilbert's initial programme.

The abstract methodand the formalist programmefascinated philosophers

It remains that Hilbert succeeded in trans-forming a philosophical question, that of foun-dations, into a mathematical problem, to bedealt with by means of the abstract methodand dependent upon a new theory, prooftheory, which remains active to this day. Inreturn, the abstract method and the forma-list programme that underlies it have exerteda kind of fascination on philosophers. Fromthe very beginning, in his Logical Researches}of 1901, then in Formal logic and transcen-dantal logic of 1929, Husserl incorporated the

abstract representation of mathematics intothe incipient phenomenology. Husserl distin-guishes two kinds of mathematics, an appliedmathematics, which includes, for example,geometry as a theory of our space, the spacein which we live, and a formal mathematics.Starting from an applied theory, a mathema-tician can extract its architecture and isolatean axiom system which he can then vary toobtain new forms for possible theories. Thus,formal mathematics appears as a theory offorms of theories or, in the language of Husserl,a formal apophantic, which aims at definingand classifying all possible systems of judge-ment. Moreover, as had been shown by Hilbert,proceeding in an axiomatic manner amountsto making abstraction of the nature of theobjects. Consequently, to each form of theorycorresponds a field of objects, arbitrary objects


Edmund Husserl (1859-1938), who was partly inspired by mathe-matical problems to develop his philosophy. (Photo AKG)

which are determined by this fact alone thatthey are subject to a certain system of axioms.The theory of forms of theories thus repre-sents a formal ontology, a theory of a pure``something'', which aims at defining and clas-sifying all possible multiplicities of objects bytheir form alone. Formal mathematics com-prises a double orientation: it is a formal apo-phantic when the mathematician turns to sys-tems of judgements; it is a formal ontologywhen the mathematician turns to fields ofobjects. If Husserl, who had closely studied19th century geometry, had the concepts ofform of theories and of formal multiplicitybefore 1901, it is certain that the meeting withHilbert and the discussions at the GöttingenMathematical Society have played a decisivepart in the development of a systematic phe-nomenology.

Hilbert succeeded in posing inside mathe-matics the question of the foundations ofmathematics. This is internalisation in mathe-matics of a philosophical question. Husserlmade the opposite internalisation, of the abs-tract mathematical method into philosophy.The careers of the two men, Husserl the phi-losopher and Hilbert the mathematician, area testimony to a reciprocal and simultaneousinternalisation, of mathematics into philoso-phy and of philosophy into mathematics.

Pierre Cassou-NoguèsCNRS, Laboratoire Savoirs et Textes,

Université Lille III

The philosopher and the mathematician 55

Some references:

• P. Cassou-Noguès, Hilbert(Les Belles Lettres, 2001).

• P. Cassou-Noguès, De l'expérience mathématique.Essai sur la philosophie des sciences de JeanCavaillès (Vrin, 2001).

• J.-T. Desanti, La philosophie silencieuse (Seuil, 1975).• D. Hilbert, Gesammelte Abhandlungen

(Springer, Berlin, 1931-35).• E. Husserl, Recherches logiques (tr. fr. H. Elie,

A. L. Kelkel et R. Schérer, P. U. F., 1959).• C. Reid, Hilbert (Springer, 1970).• H. Sinaceur, Corps et modèles (Vrin, 1991).

Jean-Jacques Laffont

Thanks to the Internet in particular, auctions have become widespread.Modelling these sales processes makes it possible to determine their

rules and the optimal strategies for using them.

auctions, as a meansof buying or selling, havebecome widespread. Thisis, in particular, the caseon the Internet, as testi-fied by the striking suc-cess of sites such as eBaywhere goods of all kinds- from books to cars,including art objects andhousehold appliances -are bid on. As a methodfor allocating scarceresources, auctions aretraditional in the live-stock markets and foragricultural produce(fish, flowers, etc). They have recently beenextended to more expensive goods, such asapartments, and to much more complexobjects, such as licences for third generationmobile telephony.The use of the system of bidding is very old,and goes back to Antiquity. Thus, Herodotus

describes the marriage market in Babylon asan auction which started with the most beau-tiful young women who were sold to thehighest bidder (i.e., the highest offer gets the`òbject'' to be sold). In Asia, the oldest accountof bidding relates to the sale of the effects ofdead monks in the 7th century.

How to rationaliseauction sales

An auction at Christie's of works of artists of the 20th century. Each potential buyer behaves accordingto what he believes the others will do. Game theory analyses such situations and helps in finding opti-mal strategies (Photo Gamma Connection/Jonathan Elderfield)

The first ideas about auctions wereinadequate because they were toosimplistic

If auctioning goes back almost to the dawnof humanity, their conceptualisation is muchmore recent. The first important academicwork devoted to the subject is a 1955 thesis,whose author was the American L. Friedman.It was one of the first theses on operationsresearch. It was devoted to developing auc-tion strategies by companies on the occasionof the sale of the petroleum drilling rights inthe Gulf of Mexico. These auctions were ̀ `first-price sealed-bid auctions'': in this procedure,the offers are not made publicand it is the highest offer whichwins the auction.

The strategy adopted byFriedman consisted simply inmaximising what is called profitexpectancy. If he wins the bid, thebidder makes a profit equal tothe difference (v - b) between hisvaluation v of the object put upfor sale and the price b which heproposes to pay for it. The profitexpectancy is thus this differencemultiplied by the probability P(b)of winning the bid at this price,that is to say (v - b) P(b). The pro-bability P(b) is a priori unknown;but by making a statistical ana-lysis of past biddings, one can dis-cover ways of outbidding thecompetitors; that makes it pos-sible to determine an approxi-mation to the function P(b) andthus find the bid b* which maxi-mises profit expectancy, i.e., suchthat (v - b*)P(b*) is maximum

This method, which is widely used and hasbeen refined in many ways, is however extre-mely na{\"\i}ve. Indeed, it implicitly supposesthat the other bidders have not worked outa strategy and that their future behaviour canbe easily deduced from their past behaviour.In 1961, the Canadian William Vickrey (whoreceived the Nobel Prize for Economics in 1996,two days before his death) posed the problemdifferently, by using game theory.

How to rationalise auction sales 57

The home page of the Internet auction site eBay-France.

Game theory and mathematical eco-nomics enter the scene for defining opti-mal strategies

Created by the famous mathematician ofHungarian origin John von Neumann in theyears 1920-1940, in collaboration with the eco-nomist of Austrian origin Oskar Morgenstern,game theory examines strategic interactionsof the players. It deals with any situation whereeach player must make a decision which deter-mines the outcome of the situation. Gametheory thus applies to many scenarios of theeconomic, political, diplomatic or militaryworld. But let us return to our biddings. Whena bidder has to decide what to bid, he askshimself what the behaviour of his competi-tors is going to be, and each bidder does this.An equilibrium of this situation indicates forthe specialists a rather complex object: it is amethod of bidding - in other words a relationbetween the valuation v of the bidder and hisbid b - which is the best way for the bidder tobid taking into account what he anticipatesare the bidding strategies of the other actorsand his guess about their valuations. Forexample, in a symmetrical situation where theexpectations of all the actors are the same,the strategy of a bidder must maximise hisprofit expectancy knowing that all the othersare using the same strategy as he is.

The concept, which we have just evoked,is a generalisation of Nash equilibrium, adap-ted to the context of incomplete informationabout the bids. What is it all about? TheAmerican mathematician John Nash (NobelPrize for Economics in 1994) had proposedaround 1950 a very natural concept of equili-brium which generalises the one given in 1838by the French mathematician and economistAntoine Cournot. Given a set of actions from

which the players can choose, these actionsform a Nash equilibrium if the action eachplayer chooses is the best possible one for him,knowing that the other players also are choo-sing the actions specified by the Nash equili-brium. In a Nash equilibrium no one finds itbeneficial to unilaterally change his action.

The particular difficulty in auctions is thateach bidder is the only one who knows hisown valuation of the goods to be sold andthat he does not know the valuations of other


The American mathematician John Forbes Nash, born in 1928,received the Nobel Prize for Economics in 1994, in particular forhis work on game theory. Around the age of thirty, Nash started suf-fering from serious mental disorders from which he made a specta-cular recovery in the middle of the 1980’s. His life was the subjectof a biography `À beautiful mind'', which inspired a film of thesame title. (Photo University of Princeton)

potential buyers. It is thus necessary to gene-ralise the concept of the Nash equilibrium tothis situation where information is incomplete.This is what was carried out intuitively byVickrey in 1961; the American of Hungarianorigin John Harsanyi did it more preciselyaround 1967-1968, which won him as well theNobel Prize in 1994. One thus arrived at thenotion of a Bayesian Nash equilibrium, aconcept of equilibrium which allows one toput forward a conjecture about the way inwhich rational bidders must bid in an auction.

In the context of auctions, a strategy fromthe mathematical point of view is a functionS which associates to a bidder's valuation hiscorresponding bid. In other words, for anyparticular valuation v, this function must spe-cify the bid b* = S(v) which maximises his pro-fit expectancy as calculated from the rules ofthe auction, supposing that the other playersuse the same strategy. That means that in asymmetric Bayesian Nash equilibrium, if theothers bid in the same way, employing thesame strategy, this way of bidding is optimal.Why have we used the adjective Bayesian?Because the player calculates his profit expec-tancy starting from his beliefs about the valua-tions of the other bidders (in probability andstatistics, the Bayesian point of view - namedafter Thomas Bayes,a British mathematicianof the 18th century - consists in evaluating pro-babilities on the basis of the available partialinformation and on a priori beliefs).

When theory confirms and extendsthe utility of sales methods arrived atintuitively...

In the auction field mathematics makes itpossible to model the behaviour of bidders,

which leads to a prediction about their wayof bidding. That has led to progress in twodirections. On the level of positive knowledge,it has become possible to compare the data,i.e., the bids of the players in different typesof auctions, with those predicted by the theory.The theory thus acquires a scientific status:one could reject it if one finds data whichcontradict the predictions; the theory is thusrefutable..

At the level of establishing standards, theconsequences have been even more impor-tant. Within the framework of the assump-tions of the theory of auctions thus construc-ted, one could prove a rather fascinatingtheorem: the revenue equivalence theorem.Without going into the details, this theoremshows that the first-price or second-price (thewinning bidder pays only the second-highestprice, not the highest price) sealed-bid auc-tions, ascending (English) or descending(Dutch) oral auctions are equivalent for theseller and they are, moreover, often optimal.Thus, sales methods which were used prag-matically in particular cases have turned outto be, in the light of theory, the optimal waysto allocate scarce resources. Hence the newenthusiasm for extending these methods toall kinds of economic activities. Finally, in morecomplex circumstances than the sale of asimple object, theory makes it possible toconceive generalisations of simple auctions inorder to optimise even more the seller'sincome, or social well-being if the organiserof the auction is a State concerned with thisaspect of things. Thanks to mathematics, ithas been possible to understand the meaningand the importance of an ancestral practiceand thereafter to transform human intuitioninto a true tool for economic development.

With the emergence of the Internet and

How to rationalise auction sales 59

the new communication technologies, auc-tions have found an immense field for expe-rimentation. The Internet offers new possibi-lities for this system of selling, which theorywill help to evaluate and to exploit. Forexample, in an auction an anonymous sellershould a priori suffer from the asymmetry ofinformation - he is the only one who knowsthe quality of the goods he is selling - andwould therefore manage to obtain only a verylow selling price; but by repeated sales of qua-lity objects to a priori unknown potentialbuyers, he can build a reputation little by little,thanks to the satisfied comments of his pre-vious buyers. The quality of the transactionscan thus be improved by creating a placewhere one can build a reputation for qualityand honesty, something to which an Internetsite lends itself easily.

Jean-Jacques LaffontInstitut d’économie industrielle,

Université des sciences sociales,Manufacture des tabacs, Toulouse


Some references:

• I. Ekeland, La théorie des jeux et ses applicationsà l’économie mathématique (P.U.F., 1974)

• A. Cournot (1838), Recherche sur les principesmathématiques de la théorie des richesses(Calmann-Lévy, Paris, rééd. 1974).

• J. Crémer et J.-J. Laffont, « Téléphoniemobile », Commentaire, 93, 81-92 (2001).

• L. Friedman, « A Competitive bidding strategy »,Operations Research, 4, 104-112 (1956).

• J. Harsanyi, « Games with incomplete informationplayed by bayesian players », Management Science,14, 159-182, 320-134, 486-502 (1967-1968).

• J.-J. Laffont, « Game theory and empirical eco-nomics : the case of auction data », EuropeanEconomic Review, 41, 1-35 (1997).

Philippe Février and Michael Visser

Great wines and Treasury bonds are sold at auction.But which type of auction should one adopt ?

To find that out, one has to supplementthe general modelling of auctions by econometric studies.

in the sale rooms ofRichelieu-Drouot, winesare sold by the traditio-nal ascending bids, orEnglish auction. In sucha procedure, the auctio-neer announces a relati-vely low starting price,and then increases it gra-dually until there remainsonly one bidder, theothers having given up.The object to be sold isacquired by this last bid-der at the highest pricereached. When severalidentical lots of wine must be auctioned, amechanism called purchase option allows thewinner of one lot in the auction to choose thenumber of lots which he wishes to acquire atthe same price (if the purchase option is notavailable, the lots are put up for auction suc-cessively). Let us consider for example the casewhen two identical lots of six bottles of

Mouton-Rotschild 1945 are to be sold. Afterthe first auction is over, the auctioneer pro-poses to the winner to purchase the second lotat the same price as the first. If the winner exer-cises his option, there is no second auction andthe two lots are sold to the winner of the firstauction. If the winner does not exercise hisright, the second lot is put up for auction.

The econometrics of sellingwines and bonds

A candlelight auction sale of wines at the Hospice in Beaune, in Burgundy. Econometric analyses indi-cate that the recourse to purchase options makes it possible to increase the income of the auctioneer.(Photo Gamma/Alexis Orand)

Purchase options, of course, make sales quic-ker, but they also introduce a strategic com-ponent. It is clear that bidders do not behavein the same way when the purchase option ispresent as they do when it is absent. In the firstcase, losing the first lot implies potentially theloss of the two lots if the winner uses his option,which is not the case in the absence of theoption. What is then the strategic impact ofpurchase options? Does the presence of thepurchase option encourage the bidders to raisetheir bids higher than when there is no option,and does it thus increase the income of the auc-tioneer?

Must the State adopt uniform auc-tions or discriminatory auctions?

The French State finances its debt byissuing bonds called Treasury Bonds. The saleof these bonds is done with the help of anauction procedure known as a discriminatoryauction. Each bidder, called a Specialist inTreasury Bonds or STB, establishes a set ofprice-quantity pairs (p,Q(p)) which determinethe number Q(p) of bondshe wishes to buy if theprice of the Treasury bondis p. The State, havingannounced the totalnumber T of bonds that itwishes to issue, the aggre-gate demand (sum of thedemands of individualbidders) determines aprice known as the equi-librium price p*: it is theprice p* such that T =Q1(p*) + Q2(p*)+... +QN(p*), where Qi is the

number of bonds the ith STB wishes to pur-chase. Each one of them then gets Qi(p*) bondsthat he had asked for.

If the price each bidder pays for a bondwere p*, this would be an auction known asuniform, and the total cost for the bidderwould then be simply p*Q(p*), the price of abond multiplied by the number of bonds requi-red (it is the area of the shaded rectangle inthe graph on the left). But in the auctionknown as discriminatory, to which the Stateresorts, the price which is paid is not p* foreach bond, but a little higher. In effect, theState makes the bidders pay the maximum ofwhat they were ready to pay for each addi-tional obligation; the total cost for the bidderis represented by the area of the shaded regionin the graph on the right-hand side.

Let us illustrate it with the example of abidder whose demand graph is as follows: heasks for 10 bonds if the price is 90 Euros, 9bonds if the price is 100 Euros,... , 1 bond ifthe price is 180 Euros. Supposing that the equi-


In uniform bidding for Treasury bonds, the bidder pays the sum of p*Q(p*) (the area of the shadedrectangle in the graph on the left), where p* is the price known as the equilibrium price of a bond,determined by the demand of all the bidders, and Q(p*) the number of bonds the bidder has deman-ded at this price. In discriminatory bidding, the bidder pays a price higher than p*Q(p*), corres-ponding to the shaded region in the graph on the right-hand side. The strategies of the bidders arenot the same in these two types of auctions.

librium price p* is 130 Euros, this bidder willreceive 6 bonds. In a discriminatory bidding,the price which he will pay is the maximalprice that he was ready to pay for these 6 obli-gations, that is to say: 180 Euros for the first,170 Euros for the second, ..., 130 Euros for thesixth, which comes to a total of 930 Euros. Inthe procedure of uniform bidding, this bid-der would have paid 130 Euros for each of thesix obligations, that is to say a total of 780Euros.

Obviously, the STBs do not bid in the samemanner in the two types of bidding and thecomparison of the two mechanisms is notobvious. Mexico changed the bidding proce-dure in 1990 in favour of discriminatory bid-ding. The United States, on the other hand,gave up discriminatory bidding in 1998 foruniform bidding. Is uniform bidding more pro-fitable for the State? Should France alsochange its mode of auction?

One has to compare two situations whe-reas there are data for only one ofthem

The answers to such questions concerningpurchase options for the sale of wine or dis-criminatory biddings for bonds are important.The amounts involved can be considerable:185 billion Euros in the year 2000 for the awardof Treasury bonds and several tens of millionEuros per annum for Drouot. How does onesolve problems of this type? The most effec-tive method would consist in carrying out areal-life experiment. Thus, in the auction ofTreasury bonds, it would be necessary to resortto the two modes of bidding in parallel tocompare the results. In the same way, for thesale of wines through auction, it would be

necessary to carry out both modes of bidding,the one with and the one without purchaseoptions, to compare the behaviour of the bid-ders. Unfortunately, it is very seldom possibleto set up such experiments. We are thusconfronted with the following problem: tocompare two situations by having informa-tion a priori only about one of them

The solution of our problem utilises a com-plex mathematical procedure. Initially, it isnecessary to model the behaviour of the bid-ders. Bidders are characterised by the maxi-mum price that they are ready to pay to obtainthe object on sale, price which one calls theirvaluation. In this model, each player knowshis own valuation, but does not know thevaluations of the other players. He only makesa guess about the valuations of the others andthis guess can be represented by a function fwhich specifies the probability with which thevarious values are taken: f(v) is the probabi-lity that the bidder allots the value v to thegoods to be sold. The optimal bidding stra-tegy, i.e., the price that he must offer accor-ding to his own valuation, is obtained by fin-ding the Bayesian Nash equilibrium (see thepreceding article, by Jean-Jacques Laffont).

One can thus model, from a theoreticalpoint of view, two concrete situations whichone wants to analyse, making it possible tocompare them theoretically. This comparisondepends obviously on the choice of the func-tion f. If, whatever the function f, one of thetwo situations dominates the the other (forexample, if discriminatory bidding allows theState to earn more income than uniform bid-ding, whatever the beliefs of the STBs as defi-ned by the function f), it is possible to arriveat a conclusion. In general, the situations tobe analysed are too complex for such domi-

The econometrics of selling wines and bonds 63

nance to occur. At this stage one gets conclu-sions of the following type: if the function fis of a certain kind, then Drouot will find itbeneficial to maintain purchase options, butif the function f is of some other kind, this isno longer the case. The problem thus comesdown to that of effectively knowing this func-tion f.

It is the comparison between real data andthe predictions of the theory which makes itpossible to determine f. Indeed, if one choosesa given function f, the model of the behaviourof the bidders and the equilibrium strategieswhich it calculates tell us what the biddersshould bid. It is then enough to compare thesepredictions - computable since we made achoice for the function f - with actual data. Ifthey coincide, it is that the chosen function fwas accurate; otherwise, it is necessary to startagain with another function.

Two types of econometric methods toassign probabilities to bidders'valuations

It is not possible in practice to test all theconceivable functions f one after the other:there is an infinity of them! To determine f,one must call upon methods called econome-tric methods. One can classify them in twomain categories: parametric methods (in whichone assumes that the function f is completelycharacterised by a certain number of unknownparameters) and non-parametric methods(which make no a priori assumptions about f).The latter ones, which are more general, butalso more complicated, were developed star-ting from the end of the 1950’s, but it is onlyvery recently that researchers have succeededin adapting them to the problem of estima-

ting our famous function f. Once this functionf is found (or, equivalently, the values of theparameters defining f in the parametricmethods have been found), it is enough tocompare the two situations being studied tofind out which one dominates the other, whichone is more advantageous from the point ofview of the seller.

Purchasing options and discrimina-tory bidding: models show thatthese procedures are advantageousfor the seller

It is this method which allowed us to ans-wer the question about using purchase optionsin wine auctions at Drouot, posed at the begin-ning of this article. We first developed twotheoretical models, one with the purchaseoption and the other without the option, andcalculated the Bayesian equilibriums in thetwo situations. We then went to Drouot toobtain data (selling price of the wines, cha-racteristics of these wines, etc.), then weapplied a parametric estimation method toour theoretical model with purchase options.It is important to note that all wines are notidentical (year, colour, chateau, level, label,etc.) and therefore it is necessary to carry outan estimate of the function f for each cate-gory of wine. Once these estimates were made,the theoretical model without purchase optionenabled us to calculate the income which theauctioneer would have had if he had not usedthe purchase option method. The first conclu-sion of this study is that the use of purchaseoptions allows the auctioneers to increase theirincome by 7% compared to the situationwithout purchase option.

Starting from data regarding the sale of


French Treasury bonds in 1995, we have imple-mented the same type of procedure for com-paring the two kinds of auction (uniform bid-ding versus discriminatory bidding). The resultsof this work show that with discriminatorybidding, the State income is 5% higher thanwhat it would have been with uniform bid-ding. Thus, in this problem as in that of wineauctions, elaborate econometric models cananswer questions which it would have see-med impossible to answer because of theabsence of data concerning one of the twosituations to be compared.

Philippe Février 1, 2 et Michael Visser 1

1CREST-LEI (Centre de recherche en économie etstatistique-Laboratoire d’économie industrielle, Paris)

2INSEE (Institut national de la statistiqueet des études économiques)

The econometrics of selling wines and bonds 65

Some references:

• C. Gouriéroux et A. Monfort, Statistique et modèleséconométriques (Economica, 1989).

• P. Février, W. Roos et M. Visser, « Etude théo-rique et empirique de l’option d’achat dans lesenchères anglaises », Document de travail duCREST (2001).

• J.-J. Laffont, H. Ossard et Q. Vuong,« Econometrics of first price auctions »,Econometrica, 63, pp. 953-980 (1995).

• S. Donald et H. Paarsch, « Piecewise pseudo-maximum likelihood estimation in empiricalmodels of auctions », International EconomicReview, 34, pp. 121-148 (1993).

• P. Février, R. Préget et M. Visser,« Econometrics of Share Auctions », Documentde travail du CREST (2001).

• E. Guerre, I. Perrigne et Q. Vuong, « Optimalnonparametric estimation of first price auc-tions », Econometrica, 68, pp. 525-574 (2000).

• W. Härdle, Applied nonparametric regression(Cambridge University Press, 1990).

Jean-Christophe Culioli

Problems of organisation and planning faced by an airline company aresimilar to those met with in other sectors of industry.

Operations research, with which tens of thousandsof mathematicians and engineers in the world concern themselves, tries

to solve these problems as well as possible.

air transport is a complex activity. It bringsinto play heavy investments (such as planesand maintenance infrastructure), highly qua-lified personnel (such as navigating personnel)and real-time expensive data-processing (reser-vation and management systems). It is also asector where competition is intense, where theposted prices do not always reflectinstantaneous production costs. Soto be competitive and safe at thesame time, an airline company mustbe managed with precision.

In order to achieve this, it has to callupon optimisation techniques speci-fic to each stage of the activity. Onegathers these mathematical tech-niques under the name of operationsresearch. This field was born underthe impetus of the military needs ofthe Anglo-Saxons during the SecondWorld War, with the beginnings ofcomputers and of methods known aslinear programming (see box).

Operations research has developed conside-rably since then and has penetrated the busi-ness and industrial world. Given the stakes, itsmethods are sometimes confidential.

Operations research is supposed to solvequestions of time scheduling, task allocation,sequencing of manufacturing stages, etc., where

Puzzlesfor airline companies

To use its fleet in the best way possible, an airline company must carefully establish itsmaintenance programme and its flight programme, plan the work of the personnel onthe ground and the rotation of crews, etc. These are difficult problems of operations research,which use equations involving several thousands of unknowns. (Photo Air France)

a number of variables and constraints are pre-sent and the solution has to be the best pos-sible one - in the sense of being cheaper, ortaking less time, etc. An elementary exampleof a problem of operations research is the oneof assigning, in a company which has 50 tasks,a specific task to each of the 50 employees,taking into account the aptitudes of each one.The best solution to this problem can, of course,be obtained by going through all the possibi-lities and evaluating each one, then choosingthe most suitable one. But this is completelyout of the question in practice: an astronomi-cal number of possibilities, namely 50! = 50 x49 x 48 x ... x 3 x 2 x 1 (approximately equal to3 x 1064), would have to be explored. Even if acomputer could go through a billion possibili-ties per second, it would take 1048 years to gothrough them all, much longer than the esti-mated age of the Universe (approximately 1010

years)!

From this example, one can imagine the

ingenuity which operations research mustbring to bear to deal with such problems in arealistic way, within an acceptable amount ofcomputing time. Moreover, various compu-tational tools and varied mathematical tech-niques (algebraic, probabilistic, numerical,etc.) enter into the design of its methods.Although it was born more than fifty yearsago, operations research is still a young science:hardly more than three years pass betweenthe moment when a new method is concei-ved in a research laboratory and the momentwhen it is applied in actual production, afterhaving passed through the engineering anddesign department. In the airline sector, thestakes are such that they have led to the crea-tion of many companies devoted to mathe-matical and computational service and consul-tation, groups like Sabre, which arose out ofthe operations research department of thecompany American Airlines, the companyAdopt, which arose from the laboratory Gerad(Groupe d’études et de recherche en analyse

Puzzles for airline companies 67

Linear programming

Linear programming is the mathematical problem which consists of determining positive quantities x1,x2, …, xN which minimise a certain ``cost'', assumed to be equal to c1x1 + c2x2 + ... + cNxN, where c1, c2,...,cN are known numbers, and the xi are in addition subject to constraints expressed by linear equations (A1x1+ A2x2 + ... + ANxN = B, where the Ai et B are known numbers which depend on the actual problem). Alarge number of questions in operations research can be formulated in these terms. If the statement of the pro-blem of linear programming is relatively simple, its solution is far from being simple, the more so as the num-ber N of unknown quantities to be determined can reach in practice several thousands. This apparently tri-vial problem, but of capital importance for its applications, is at the origin of the most significant researchin optimisation for the last thirty years. In 1947, the American mathematician George Dantzig put forwardthe excellent simplex algorithm which is still frequently used. In the 1970’s and 1980’s, other competing algo-rithms made their appearance. The year 1984 marked a turning point: a young mathematician working inthe United States, Narendra Karmarkar, discovered a particularly efficient linear programming algorithm(having so-called polynomial convergence). The ideas underlying his method initiated a active wave of research(method of interior points) pursued simultaneously by thousands of mathematicians in the world. Thanks totheir efforts, industry now has a palette of very powerful linear programming algorithms.

des décisions) of the University of Montreal,or French companies like Eurodecision, Ilog,or Cosytech.

Optimal programming of flights,allotting an aircraft to each flight,minimising the downtimes

To use its fleet of aircraft - the greatestasset of any airline company - as efficiently aspossible, it is necessary to start by establishingan optimal maintenance programme, by fixingthe times of the minor and the major techni-cal inspections that each plane must undergo.As a grounded plane does not brings in anyrevenue, one must minimise the downtime ofeach aircraft by taking into account the sche-dules and the qualifications of the agents, theavailability of hangars, etc. The equations towhich the problem leads are not linear.Therefore they present some difficulties, butsufficiently effective methods to treat themhave been developed of late.

Once the maintenance programme (for aperiod of 6 months to 10 years) has been esta-blished, one must establish an optimised flightprogramme. After having constructed a net-

work - a list of routes to be operated alongwith the associated schedules, according tothe forecasted market share and the windowsallotted to each company by the IATA(International Airline TransportationAssociation) - one determines which type ofaircraft (Airbus 340, for example) will be bestadapted, technically and economically, for car-rying out each of these flights. The data whichenter into the optimisation programmes arethe characteristics of the plane (capacity, per-formance), estimated number of passengers,etc. Developing the programme of flightsrequires optimisation techniques calling uponstatistics and probability theory, as well as inte-ger linear programming algorithms (wherethe unknowns represent integers).

It is then a question of arranging the flightsand the maintenance operations of each planeso as to satisfy all the operational constraints(authorised successions or otherwise, mainte-nance rules, etc.), while minimising the pos-sible consequences of technical breakdowns,unforeseen delays, etc. This optimisation pro-blem, known as aircraft schedule construction,is modelled as a very large integer linear pro-gramming problem. To be solved exactly, itrequires the application of a decompositiontechnique (column generation, Lagrangianrelaxation).

Finally, for each aircraft schedule, it isnecessary to determine which particular planewill be assigned to it, in accordance with themaintenance constraints of each aircraft (thenumber of hours of flight, the number of lan-ding/takeoff cycles before each maintenanceinspection, etc.). This matriculation is gene-rally achieved by a method of ``dynamic pro-gramming''. Introduced by the AmericanRichard Bellman in the 1950’s, this step consistsof breaking up the initial decision problem


Airline companies seek to reduce as much as possible the time theiraircraft have to remain on the ground, while taking account of mul-tiple constraints: a plane on the ground does not bring in any reve-nue. (Photo Air France)

Puzzles for airline companies 69

into several simpler problems which can besolved one after the other (dynamic pro-gramming can apply equally well to the cal-culation of the optimal trajectories of theplanes and to the determination of financialinvestment strategies).

Planning problems can take onvarious forms, and so does theunderlying mathematics

As each plane has a precise programmedetermined in advance, one can then try tomaximise its expected revenue by opening orclosing reservation for various classes accor-ding to the likely demand by customers. Thisis a very classic problem in aviation, in thetransport of passengers by rail, in car-hire firmsand in hotel chains. It is a stochastic optimi-sation problem where it is necessary to maxi-mise the revenue F in the probabilistic sense,i.e., to maximise the expected revenue F kno-wing that F depends on random variables xi

(the xi can, for example, stand for the num-ber of passengers in each of the classes, withconstraints of the form A1x1 + A2x2 + ... + ANxN

= B, where B represents a capacity).

It is necessary to add, to all that precedes,planning for ground personnel (size of themanpower, synchronisation with the flight pro-gramme, taking charge of the transit passen-gers and their luggage, etc.) and for flightcrews, of course taking into account work regu-lations and safety norms. As one sees, the acti-vities of an airline company pose a large varietyof optimisation problems, which are often simi-lar to those of rail or maritime transport. Theseproblems are difficult; mathematically, theycorrespond to the minimisation or the maxi-misation of quantities depending on a large

number of variables (often several thousandsor more). Nevertheless, efforts of operationsresearch have borne fruit and today one hasvery good algorithms for a vast majority ofsituations. But nobody in this field can rest onhis laurels: research must continue, as the per-formance of the company depends on it.

Jean-Christophe CulioliDirecteur de la recherche opérationnelle

Air France

Some references:

• Y. Nobert, R. Ouellet et R. Parent, La rechercheopérationnelle (3e éd., Gaëtan Morin, 2001).

• R. Faure, B. Lemaire et C. Picouleau, Précis derecherche opérationnelle (5e éd., Dunod, 2000).

• « AirWorthy OR » dans Operational Researchand Management Science Today, numéro dedécembre 1999.

• Bulletins de la ROADEF (Association pour laRecherche Opérationnelle et l’Aide à laDécision en France, issue de la refondation

de l’AFCET).

Maurice Mashaal

Physicists have aspired for a long time to building a theory which wouldcover all the elementary particles and all their interactions.

Since about twenty years ago, they have a trail that is promising.But to explore it, they must navigate in highly

abstract spaces where even mathematicians had not yet ventured

every reasonable person knows that scien-tists such as physicists or chemists make use ofmathematics. But few know to what extentthis is so, and how deep the interactions bet-ween mathematics and the natural sciencesis. Galileo said that the book of Nature is writ-ten in the language of mathematics. This ideais fully confirmed by the modern developmentof science and particularly that of physics.There is more to it than mere confirmation:many thinkers are astonished to note thatmathematical discoveries or inventions alwaysend up being used for the description of someaspect of natural phenomena. It is the asto-nishment at the `ùnreasonable effectivenessof mathematics in the natural sciences'' ofwhich the physicist of Hungarian origin EugeneP. Wigner (1902-1995) spoke.

One doesn't really know why mathematics isso `èffective''. This question, which belongsto the philosophy of knowledge, is still open.We will not try to answer it here, but merelyillustrate this effectiveness in the field of the

most theoretical and the most fundamentalphysics, which a priori does not have any mate-rial utility - but which has nevertheless givenus such crucial inventions as the laser, the tran-sistor or nuclear energy...

Physics and mathematics, a long his-tory of mutual contributions

The links between mathematics and phy-sics are not recent. Isn't the Archimedes' prin-ciple (à body floating in a liquid undergoesan upthrust equal to the weight of the volumeof the displaced liquid') a mathematical sta-tement about a physical phenomenon? Didphysics not have a spectacular period ofgrowth thanks to the creation of infinitesimalcalculus by Newton and Leibniz in the 17th cen-tury? What's more, these bonds do not alwaysgo in one direction, mathematical tools beinginvented first and then being applied to pro-blems in physics. To mention just one exampleamong many: it is while studying the problem

Geometry in 11dimensions tounderstand Genesis ?

of heat propagation thatthe French mathematicianJean-Baptiste JosephFourier (1768-1830) concei-ved his ``Fourier series''(which are infinite sums oftrigonometric functions),which have since thenplayed an extremely impor-tant role in science andtechnology.

The Physics of the 20th

century is rich in interactionswith mathematics. This isthe case with the two greattheories born at the begin-ning of this century, therelativity theory of Einsteinand quantum mechanics.Einstein's (general) relati-vity is a theory of gravita-tion which supplants that ofNewton; it rests on radicallydifferent concepts, relatedto the non-Euclidean geometries introducedin the 19th century when nobody suspectedthat such mathematics could have any rela-tionship to reality.

In the same way, when mathematiciansstarted studying ``Hilbert spaces'' (abstractspaces whose points can be - for example -functions satisfying certain technical condi-tions), at the beginning of the 1900’s, nobodysuspected that no more than only twenty yearslater, the mathematics of Hilbert spaces wasgoing to constitute the right framework forformulating the laws of quantum mechanics(which manifest themselves especially at theatomic and subatomic scales).

Conversely, fundamental research in gene-ral relativity or in quantum mechanics has in

its turn stimulated purely mathematicalresearch.

The physics of elementary particles, afield where very abstract mathema-tics is deployed

Let us look a little more closely at one ofthe ways in which quantum physics has deve-loped: the study of elementary particles andof their interactions. In the course of thedecades 1930-1950, a theoretical frameworkcalled quantum field theory was worked out;it is extremely complicated, both from thepoint of view of the concepts as of the mathe-matical techniques involved. It is within this

Geometry in 11 dimensions... 71

A myriad of very remote galaxies viewed through the space telescope Hubble. Gravitation being a keyelement of the birth and the evolution of the Universe, specialists in cosmology would like to finallyhave a description of the gravitational force compatible with the principles of quantum physics. Willstring theory grant this wish? (Photo R. Williams/HDF (STSci) /NASA)

framework, and with the discovery of newparticles thanks to particle accelerators, thatphysicists found out that the world of ele-mentary particles possesses a certain numberof symmetries. The theory of groups, an impor-tant branch of mathematics founded in the19th century, played and continues to play acrucial role in the elucidation of these (for themost part abstract) symmetries. It is thanks toit that, on several occasions, theorists havebeen able to predict the existence of certainparticles a few years before they were actuallydiscovered by the experimentalists.

In the years 1970-1980, the theory of ele-mentary particles had arrived at the pointwhere it was able to describe in a satisfac-tory and unified manner all known particlesand almost all their interactions. Why`àlmost''? Four fundamental interactionsare known - the gravitational force, theelectromagnetic force and two forces actingon the nuclear scale, the weak interactionand the strong interaction; however, physi-cists did not succeed in incorporating thegravitational force into their theory calledthe standard model of particle physics.

To reconcile gravitation with quan-tum physics: a fundamental challengewhich seems to be within reach ofstring theory

What does this exception mean?Gravitation seems to be correctly describedby the general relativity of Einstein, butEinstein's theory is not a quantum theory,i.e., it does not incorporate the principles ofquantum physics (which are quite strange,let it be said in passing). However, one doesnot at all see why gravitation should beexempt from the quantum laws which therest of Nature obeys. Hence the obstinacy oftheoretical physicists to bringing gravitationinto the quantum fold. In spite of severaldecades of effort, they have not succeededin doing so.

However, since the middle of the 1980’s,many of them believe that they have foundthe right track. Indeed, it was at that time thata new theory, called the theory of strings,which is still unfinished but holds a lot of pro-mise, gained sufficient coherence for it to betaken seriously. The exact context and the pre-cise reasons which pushed theorists in thisdirection are too technical to be explainedhere. It is also impossible to explain in a simpleway what string theory is. Let us just roughlysay that it posits that the fundamental physi-cal objects are not point particles (the under-lying ``philosophy'' of the traditional quan-tum field theories), but tiny strings withoutthickness - small portion of a line of some sort;and that the various particles observed at ourscale correspond to different vibrational modesof these strings, a little like the various musi-cal notes which are nothing but different vibra-tional modes of a violin string.


A vibrating closed string has a whole number of peaks and troughs.Various subatomic particles (electrons, photons, etc.) would corres-pond to the various vibrational modes of a tiny fundamental string.

For string theories to be consistent, itis necessary that space-time have 11dimensions

String theories (theories in the pluralbecause there are in fact several variants) arestill preliminary and formidably complex. Anumber of their aspects remain to be clarified.Moreover, for the moment it is impossible totest them experimentally because the ener-gies that would be required are completelyinaccessible, even with the most powerful par-ticle accelerators that we have. But they arealluring to theorists because these (quantum)theories incorporate gravitation in a naturalway, apparently without running up againstthe obstacles which surfaced in the earlier theo-ries.

If physicists succeed in constructing a com-plete and consistent theory of strings, theywill be able to make a precise study of violent(i.e., of very high energy) gravitational phe-nomena which take place in the cosmos, suchas the collapse of a massive star onto itself,the physics of ̀ `black holes'', etc. Also, one willbe able to understand better the mysteries ofthe very first moments of the birth of theUniverse - the first moments of the famousBig Bang, one of the most violent events. Aquantum description of gravitation will cer-

tainly render it possible to make a qualitativeand quantitative leap in our understandingof the Universe, of its origin, and of its evo-lution.

But, as was said above, string theories arevery complicated. They involve very sophisti-cated mathematical techniques, often the pro-duct of the most recent research. In fact, thespecialists who study these theories includeboth physicists and mathematicians (severalwinners of the Fields medal, the highest awardin mathematics, have devoted a significantpart of their work to string theories; such isthe case of the American Edward Witten, orof the Russian living in France MaximKontsevitch). It was, in particular, establishedthat string theories cannot be consistent unlesswe take space-time to have not four dimen-sions (three dimensions of space, one dimen-sion of time), but many more: 11 dimensionsat the last count! These seven additionaldimensions, imperceptible to our sensesbecause they are supposed to be wrapped upover themselves in tiny loops, contribute tothe abstraction and the difficulty. The theo-rists' need to handle strings and other objectsin spaces having such a number of dimensionscreated an excellent ground for collaborationbetween physicists and mathematicians.

Research in this field has benefitedstring theory as much as variousbranches of fundamental mathe-matics. It is a beautiful example, inthe history of physics and mathe-matics, of an intimate connectionbetween these two disciplines,research in one field fed by theresults of the other. But the gameis well worth it: although stringtheories are still highly speculative,it is a question of nothing less than

Geometry in 11 dimensions... 73

Diagrammatic representation of the interaction between two strings. In the course oftime, which flows from left to right in this diagram, a closed string sweeps out a sur-face similar to a tube.

penetrating the enigmas of the infinitely smalland the infinitely large, i.e., ultimately, thequestion of our origins.

Maurice MashaalScience journalist


Edward Witten, one of the principal craftsmen of string theory. Onedoesn't know if he should be considered a physicist or a mathemati-cian... (Photo: DR.)

Some references:

• B. Greene, The Elegant Universe (Norton, 1999)• P. Deligne et al. (eds.), Quantum fields and

strings : a course for mathematicians (AmericanMathematical Society/Institute for AdvancedStudy, 1999).

Internet: modellingits traffic for managingit better

François Baccelli

Specialists in communication networks tryto understand the statistical properties of the data traffic

which they have to route. The management and the developmentof these networks depend upon it.

communicationnetworks (tele-phone, Internet,local area net-works, etc.) haveundergone a phe-nomenal expansionin the last fewdecades. For theiroperators, a centralquestion is kno-wing how tocontrol the flow ofinformation in anoptimal way, inorder to avoid any bottlenecks and to offerusers a service of good quality which is fastand reliable. A thorough knowledge of theproperties of the communications traffic insuch networks is essential for the conceptionof effective information flow control proce-dures or the correct design of the necessarysoftware and hardware equipment.

The mathematical analysis of traffic in com-munication networks is already an old subject.It goes back to 1917, with the work of the Danishengineer Agner K. Erlang. His approach, conti-nued by many other researchers, provided themain mathematical tools for designing used bythe operators and the manufacturers of net-works until the year 1990 or thereabouts.

The Internet is not a centralised network as the previous communication networks were. Such structuralchanges have major effects on the mathematical properties of the data traffic. (Photograph: Getty Images)

Until the 1990’s, modeling the trafficby traditional statistical laws wasadequate

The mathematical procedure explored byErlang and other researchers and engineersafter him is Markovian in principle. This meansthat the description of the traffic is basedupon a simple model of random processes,called Markov chains, for which the mathe-matical theory is quite advanced and power-ful (Andreï Markov (1856-1922) was a Russianmathematician who made important contri-butions to probability theory).

In simple terms, a Markov chain is a suc-cession of random events in which the pro-bability of a given event depends only on theimmediately preceding event. In the frame-work of communication networks, theMarkovian approach of Erlang assumes thatthe statistical laws characterising the trafficare Poisson distributions; the Poisson distri-bution, named after the French mathemati-cian Denis Poisson (1781-1840), is one of themost widespread and simplest laws of pro-bability and statistics.

The Poissonian assumption proved itselfto be justified for telephone traffic (wherethe random events are calls by the subscri-bers, which occur at random times and ofwhich the duration is also random).

This kind of traffic model permitted thesetting up of adapted control procedures.Until quite recently, the control of the com-munication networks was a control of admis-sion, i.e., the operator would refuse the useraccess to the network when he could not gua-rantee a preassigned quality of service.

This type of control requires a rather pre-cise knowledge of the state of the network asa whole, and it is thus applicable only to net-

works which are managed in a centralisedmanner.

But today's communication networks arenot the same as yesterday's. Internet has under-gone an extraordinary development in the lastfive years (it is estimated that the voice com-munications traffic accounted for 90% of thetotal traffic in 1997, it was 50% in 2000 andit will represent only 10% within a year ortwo).

This growth has radically changed a situa-tion which was stable for more than half acentury. The main reason for this rapid deve-lopment lies in the use, for routing informa-tion and controlling traffic, of new decentra-lised protocols such as IP, or Internet Protocolfor routing, and TCP, or Transmission ControlProtocol for control, which make the Internetan indefinitely extensible network.

The statistical properties of the traf-fic changed. It was necessary tounderstand how and why

These structural modifications had effectson the traffic and its statistical properties, andone had to develop a mathematical theoryadapted to the new situation. Indeed, statis-tical analyses carried out in the middle of the1990’s by researchers at Bellcore in the UnitedStates and at INRIA (Institut national derecherche en informatique et en automatique)in France showed, first for local area networksand then for the Web, that the traffic couldno longer be described using Poisson's proba-bility distribution.

In particular, long memory random pro-cesses (where the probability of an eventdepends on events which happened relativelyfar back in the past) were observed, which


excludes the usual modelling based on tradi-tional Markovian processes. Often these pro-cesses also display statistical properties knownunder the name of multi-fractality, whichmeans a great irregularity. It so happens thatall these statistical properties have importantconsequences, for example, for dimensioningof router memories; not taking them intoaccount could result in underestimating theloss of information packets by the network,leading to dysfunction.

Since the first articles putting forward the

new statistical properties of data traffic appea-red, many papers have been published in orderto explain them. Today, the origin of the longmemory phenomenon observed in traffic sta-

tistics is rather well understood. One has beenable to establish that it arises directly fromthe statistical distribution of the size of thefiles stored in Web and FTP (File TransferProtocol) servers, as well as, the size distribu-tion of the files requested by users at the timeof HTTP (HyperText Transfer Protocol, usedwhen surfing the Web) and FTP requests. Theirstatistical graph, i.e. the curve representingthe number of files consulted or exchangedaccording to their size, decreases less rapidlythan the exponential on both sides of themaximum: their law of distribution is said tobe sub-exponential. It has been shown thatthe sub-exponential statistical laws obeyed bythe individual behavior of the Net surfers,superposed a large number of times - giventhe large number of Net users - leads to, as adirect consequence, the phenomenon of thelong memory characterising global traffic.

Analysing the TCP protocol and itseffects in order to improve the mana-gement of the Internet

This does not clear up everything. Currentwork concentrates on explaining statisticalproperties of the traffic on small time scales,multi-fractality in particular. The most wides-pread hypothesis is that this property resultsfrom the control protocols being used, and inparticular the TCP. But what is this TCP pro-tocol, which currently controls nearly 90% ofthe traffic on the Internet? It is an adaptiveflow control system, where the amount ofinformation emitted by a source is governedby an algorithm which increases the flow ofemission linearly in the course of time as longas no bottlenecks occur; but as soon as lossesare detected, the algorithm halves the flowof emission.

Internet: modelling its traffic... 77

Net surfers in a cybercafé. A good knowledge of the statistical pro-perties of data flow in the Internet network is essential for the Webto work smoothly. (Photo Frank Moehle)

It is this adaptive control which regulatesevery response to congestion in the network.Its mathematical analysis presents many diffi-culties because of the decentralised, stochas-tic (bottlenecks and the losses occur randomly),non-linear (the effects are not directly pro-portional to the cause), complex (very exten-sive network involving interactions betweenmany intermediate routers) character of thesituation. It so happens that developing modelsintegrating all these elements is a major stake,whether it is a question of defining the rulesfor dimensioning the network or optimisingthe flows or predicting and controlling therandom variations in the quality of the serviceprovided by the network to its users.

Scientific challenges and economicstakes which spur academics andindustrialists

Such a task requires the efforts of scien-tists in very diverse fields (statistics, the theoryof probability and queuing theory, non-linearadaptive control systems, the theory of largestochastic networks, dynamical systems) whichgo beyond those of the traditional approach.In the last few years, a large number ofmodels, some of them simpler than theothers, have been proposed. Some of themaccount for the multi-fractality of the globaltraffic, a property mentioned above; othersmake it possible to evaluate if the sharing ofthe same communication channel betweenseveral data flows controlled by TCP is equi-table, etc.

Current research also concentrates a loton the analysis of DiffServ, a differentiationmethod for the services on offer based on thecreation of priority classes for the exchange

of data. This appears to be the only extensiblemethod capable of improving the quality ofservice provided by the Internet. Anotherimportant direction concerns the adaptationof UDP (User Datagram Protocol), a protocolused for the transmission of video and voicedata, which are not controlled by TCP, in par-ticular with the aim of defining transmissionmodes for these flows which would be com-patible with TCP.

Faced with these questions, which presentscientific challenges and economic stakes ofprimary importance, the academic world andthe industrial world are organising themselves.How?

Most of the big industrial groups in infor-mation technology and the network opera-tors have formed high-level research teamswhich have been assigned the job of model-ling the traffic and control in data networks,particularly the Internet. There is an equaleffort on the part of the academic world, inparticular in the United States, in Europe andin some Asian countries, where interdiscipli-nary collaborations have been set up betweenmathematicians and computer scientists orelectrical engineers.

The agency which has the greatestinfluence on the evolution of the Internet iswithout doubt the IETF (Internet EngineeringTask Force, cf. the address http://www.ietf.org).It is open to every one, whether he is a net-work designer, researcher or operator. Its acti-vities are conducted in the form of workinggroups dealing with several fields such as rou-ting, security, transport, congestion control,applications, etc. These working groups havebeen entrusted with making recommenda-tions, some of which will become standards.Validating these recommendations by mathe-


matical studies of the type evoked in thisarticle constitutes an important component,and sometimes the decisive component, ofthe work of standardisation.

François BaccelliINRIA (Institut national de recherche en

informatique et automatique) etÉcole Normale Supérieure

(Département d’informatique), Paris

Internet: modelling its traffic... 79

Some references:

• K. Park et W. Willinger (eds.), Self similar trafficanalysis and performance evaluation (Wiley, 2000).

• P. Abry, P. Flandrin, M. S. Taqqu et D. Veitch,« Wavelet for the analysis, estimation and synthe-sis on scaling data », dans la référence ci-dessus.

• F. P. Kelly, A. K. Maulloo et D.K.H. Tan, « Ratecontrol in communication networks : shadowprices, proportional fairness and stability »,Journal of the Operational Research Society, 49,pp. 237-252 (1998).

• R. Riedi et J. Levy-Vehel, « Fractional brownianmotion and data traffic modeling : the otherend of the spectrum », Fractals in Engineering(Springer-Verlag, 1997).

• M. Taqqu, W. Willinger et R. Sherman, « Proofof a fundamental result in self similar trafficmodeling », Computer Communication Review,27, pp. 5-23 (1997).

• F. Baccelli et D. Hong, Interaction of TCP flowsas billiards, rapport INRIA, avril 2002.

Financialoptions pricing

Elyès Jouini

The financial world fixes the price of options by means of formulaswhich have been obtained thanks to relatively recent work

in mathematics. The search for better formulas continues...and small-time speculators are not the only ones who are interested!

iIn the preface to the fourth edi-tion of his Éléments d'économie poli-tique pure ou théorie de la richessesocial, published in Lausanne in1900, Léon Walras wrote: `Àll thistheory is a mathematical theory, i.e.although it can be explained usingordinary language, the proof mustbe provided mathematically''. A littlelater, he even added: ̀ Ìt is now quitecertain that political economy is, likeastronomy, like mechanics, an expe-rimental and a rational science atthe same time...The 20th century, which is not far,will feel the [...] need to place thesocial sciences in the hands of edu-cated men, accustomed to handling inductionand deduction, reasoning and experiment, atthe same time.Mathematical economics will then take its placealong with astronomy and mathematicalmechanics''..

I would like to show, through the followingexample which is borrowed from the world offinance, how mathematics and economics conti-nue to maintain extremely close links and thatthere is a real cross-fertilisation between the sub-jects of current interest in each discipline.

New York Stock Exchange on a lucky day. Mathematics has had a great influence onthe world of finance for more than twenty years. Reciprocally, the financial world pro-vides problems which stimulate research in certain fields of mathematics. (Photo GammaConnection/Gifford)

The problem which will interest us here isthe valuation of financial options. The ques-tion is as old as the options themselves, ofwhich one can find a trace, for example, inAntiquity and in the 17th-century tulip marketin the Netherlands.

It was however only in 1973, as we shallsee later, that this question found its firstmathematically satisfactory answer. And it isnot a coincidence that the first organisedoptions market, the one in Chicago, madegreat strides, which have not abated since, inthe same year

What is a financial option? Let us consi-der a certain asset quoted on financial mar-kets at the price S today. By means of anoption, financial markets offer potentialbuyers the possibility of buying this asset at alater date, let us say in three months, at theprice K. This may interest, for example, a buyerwho does not yet have the necessary fundsand who wants to guard himself against a risein the price of the asset.

Such an option is a kind of insurancecontract which confers the right to buy theasset at a later date at the guaranteed priceK. Obviously, this right itself has to be sold ata certain price, but at what price? Such is thequestion of the valuation of the option price.To speak in financial terms: what must be theprice of an option on the asset S, of ``strikeprice’’ K and maturity of three months?

It is clear that the buyer of such a rightwill exercise it only if the price of the asset inthe market is higher than K in three months.He will then be able to buy the asset at theprice K and resell it at the current price, the-reby making a profit equal to the difference.In three months, the buyer of the optionmakes a profit equal to the difference bet-

ween the current price of the asset and K, ifthis difference is positive, and none otherwise.

The principle of non-arbitrage is thebasis for determining the price offinancial assets

To fix the price of such an option, thetheory of arbitrage relies upon a very simple,indeed simplistic, principle: the absence ofarbitrage opportunities. In other words, thisprinciple affirms that it is not possible to gua-rantee, come what may, a positive profit at alater date by making zero investment today(nothing comes from nothing). The principleof the absence of arbitrage opportunities doesnot mean that miraculous profits are impos-sible. Indeed, I could very well borrow themoney to buy a lottery ticket - my personalcontribution is thus zero - then go on to wina million Euros, refund the loan and make anenormous profit. The principle only states thatsuch an outcome could not have been gua-ranteed a priori. Indeed, in the precedingexample, I could very well win nothing andbe obliged to refund my loan: I have thustaken the risk of suffering a loss.

Thus, the absence of arbitrage opportu-nities means quite simply that any profit higherthan the yield of a riskless asset (interest rates,bonds, Treasury bonds, etc.) is necessarily rela-ted to some risk. The SICAVs, for example,have an average yield higher than that of themonetary market; however, this yield is notguaranteed and may very well pass below thatof the money market, as we saw it in the year2001.

To simplify the problem, let us supposenow that the market functions only on two

Financial options pricing 81

days, today and in three months, and that theprice S of the asset in three months can takeonly two values, namely 100 Euros and 150Euros. Let us suppose moreover that K, thepurchase price agreed upon for the asset afterthe expiry of the option, lies between the highvalue of Sh = 150 Euros and the low value ofSb = 100 Euros, for example K = 140 Euros. Ifthe price of the asset in three months is thehigher value 150 Euros, the holder of theoption exercises his right and buys it at theprice of K = 140 Euros agreed upon before-hand; the profit associated to the option isthus Sh - K = 150 - 140 = 10 Euros. If the priceof the asset in three months is the lower price100 Euros, the holder of the option does notexercise his purchasing right at the price K,which is higher; the profit associated with theoption is, in this case, none.

One can show that such a profit can alsoto be made by constituting a portfolio com-prising only assets and investments (or loans)at the interest rate prevalent in the market,which we will denote here by r. Let C be thecost of constituting such a portfolio. Two assetswith identical yields have to have the sameprice (it can be proved that otherwise the prin-ciple of non-arbitrage would be violated), oneconcludes that the price of the option must beequal to C.

The cost C of the portfolio, equal to theprice of the option, can be determined in aprecise way. One shows that C is a weightedaverage of the discounted option payments,that is to say a weighted average of theamounts (Sh - K)/(1 + r) and 0/(1 + r) = 0, andthat the weights appearing in this average aresuch that the price S of the asset today is itselfa weighted average of the discounted assetpayments (Sh/(1 + r) and Sb/(1 + r)), with the

same weights. More precisely, one proves thatthere is a probability such that the price of anyasset is equal to the expectation, calculatedaccording to this law, of its discounted futurepayments. This result is deduced from ele-mentary linear algebra and is related to thesimple model presented above. But thanks tothe techniques of convex analysis, which weredeveloped in the middle of the 20th century, itcan be extended to the case where the assetprice can take several different values (finitein number).

Stochastic calculus: when financeand mathematics enrich each other

However, if one wishes to adhere moreclosely to reality and considers a model inwhich time is continuous and the asset pricehas a continuum of possibilities, it is necessary,to interpret the simple principle of arbitrage,to appeal to more advanced concepts of pro-bability theory which were developed in thesecond half of the 20th century. More preci-sely, one appeals to the theory of stochasticprocesses (processes in which quantities evolverandomly in the course of time) and the theoryof stochastic differential equations (differen-tial equations involving random quantities).The most recent developments in these fieldsare closely related to problems encounteredin finance

These models assume that the price of aasset evolves with a deterministic (non-ran-dom) yield, to which is added a random termwith zero mean and amplitude specific to theasset under consideration. This random fluc-tuation is called volatility and can depend ontime and on many other endogenous or exo-genous events. With these assumptions, one


finds that the price of the option obeys a cer-tain partial differential equation (differentialequation where the unknown functiondepends on several variables). In the simplestcase, studied in 1973 independently by theAmericans Fischer Black and Myron Scholeson the one hand and Robert Merton on theother, this equation is the same as the equa-tion of heat propagation, well-known to phy-sicists. It is then possible to solve it explicitlyand to determine the price of the option accor-ding to its own characteristics (expiry, price ofexercise) as well as the the asset price and itsvolatility: this is the formula of Black-Scholesand Merton, for which Scholes and Mertongot the Nobel Prize in Economics in 1997 (Blackdied in 1995).

This formula and its variants are used in allthe money markets of the world. Programmedon all the computers of the market rooms, uti-lised innumerable times every day, they are thevery example of the possible relationship bet-ween theoretical mathematics and concreteapplications. The formula of Black-Scholes andMerton corresponds however to the simplisticcase where interest rates, average yield rates,risk levels, etc., remain constant in the course oftime. As soon as we modify these assumptions,the equations we obtain are no longer equiva-lent to that of heat propagation. The relevantequations are a variant of this and they requirespecific methods for their solution - implicit,explicit or numerical. It is for working on someof these equations that the French researchersC. Daher and M. Romano, who were workingat that time at the Université Paris-Dauphineand at the Caisse autonome de refinancement,obtained in 1990 the IBM prize for intensivenumerical calculation.

Finally, if one tries to be even more realisticby taking into account transactions costs, various

constraints imposed by the market or the impactof the number of transactions on the price, thetraditional techniques of stochastic calculus areno longer sufficient. It is necessary to develop,as has been done in the last few years, specifictools like retrograde stochastic differential equa-tions or the fine duality methods in optimal sto-chastic control. One then discovers, and this maycome as a surprise, that these new mathemati-cal ideas, developed to solve economic and finan-cial problems, turn out to be related to problemsthat have been already encountered in geo-metry or in physics - for example the deforma-tion of surfaces or the melting of ice cubes - andthat they throw a new light on them.

Elyès JouiniProfesseur des Universités, CEREMADE (Centre

de mathématiques de la décision)Université Paris-Dauphine (Paris 9)

Financial options pricing 83

Some references:

• F. Black et M. Scholes, « The pricing of optionsand corporate liabilities », Journal of PoliticalEconomy, 81, pp. 637-654 (1973).

• C. Huang et R. Litzenberger, Foundations forfinancial economics (North-Holland, 1988).

• L. Walras, Éléments d'économie politique pure outhéorie de la richesse sociale (Corbaz, Lausanne,1874, édition définitive revue et augmentée parl'auteur, LGDJ, Paris, 1952).

• Options, Futures, and Other Derivatives (5th

Edition by John C. Hull Prentice Hall• A Random Walk Down Wall Street by Burton G.

Malkiel WW Norton & Cie NY NY• Introduction to Stochastic Calculus Applied to

Finance, Second Edition (Chapman & Hall/CrcFinancial Mathematics Series) by DamienLamberton and Bernard Lapeyre

Gilles Lachaud

For detecting and correcting the inevitable errors which creep in duringdigital information transmission, specialists in coding theory make use

of abstract methods which arise from algebra or geometry.

we are in the digital era. What does thatmean? Quite simply that an enormous part ofthe information exchanged across the planetis actually represented in the form of num-bers. E-mails, mobile telephony, banking tran-sactions, remote control of satellites, imagetransmission, CDs or DVDs, etc.: in all theseexamples, information is translated - one saysthat it is encoded (not to be confused withencrypted) - into a sequence of integers, andcorresponds physically to electric or some othersignals. Even more precisely, information isgenerally encoded in the form of sequencesof the binary digits 0 and 1, also called bits.For example, in the ASCII (American StandardCode for Information Interchange) code usedby microcomputers, capital A is encoded bythe byte (sequence of 8 bits) 01000001, capi-tal B by 01000010, etc.

A major problem of information trans-mission is that of errors. A small scratch on adisc, or a disturbance of the equipment, orany parasitic phenomenon is enough for the

transmitted message to contain errors, i.e.,``0''swhich have been inadvertently changed into``1''s, or vice versa. However, one of the manyadvantages of digitising is the possibility ofdetecting and even correcting such errors!

Communicatingwithout errors:error-correcting codes

The Matrimandir géode built by the French architect Roger Angerin Auroville (Tamil Nadu, India). In the construction of efficienterror-correcting codes, one encounters problems related to difficultquestions of pure geometry, such as how to cover a sphere by the grea-test possible number of discs of the same size without any overlaps.

The words of a message are lengthe-ned, so that after deterioration onecan still recognise them

Such is the function of error-correctingcodes, which were first designed more thanfifty years ago, and at the same time as thefirst computers. How do they work? The prin-ciple is the following: one lengthens the digi-tal ``words'' which make up the message sothat some of the bits serve as control bits.For example, in the ASCII code mentionedabove, one of the eight bits is a control bit:it must be 0 if the number of ̀ `1''s in the otherseven bits is even, and 1 otherwise. If thevalue of one of the eight bits gets acciden-tally changed, the parity indicated by thecontrol bit is no longer valid and an error isthus detected. The same idea is to be foundin many numbers which we come across ineveryday life. For example, in bank accountnumbers, a letter-key is added to the num-ber in order to be able to detect a transmis-sion error. In the same way, the numbers onEuro banknotes are encoded so as to avoidcounterfeits. In other words, the philosophyof error-correcting codes is to compose redun-dant messages: each word of the message islengthened in such a way that it containsinformation about the message itself!

A not very realistic, but simple and illu-minating example of an error-correcting codeis triple repetition: each bit of the message tobe encoded is tripled, i.e., 0 becomes 000 and1 becomes 111. This code makes it possible todetect and to correct one possible error in atriplet. Indeed, if we receive, let us say thesequence 101, then we can immediatelydeduce that the intended sequence is 111 (sup-posing that only one bit out of the three iserroneous) and thus that the initial informa-

tion was the bit 1. The triple repetition codeis not realistic because it is expensive: for eachbit of information, it is necessary to send threebits; we say that its profitability rate is 1/3.This rate has a direct bearing on the durationnecessary for the transmission of messagesand on the cost of communication.

A good error-correcting code must haveother qualities in addition to a high profita-bility rate. It also needs to have a good capa-city for detecting and correcting errors, andthe decoding procedure must be sufficientlysimple and fast. This is the whole problem ofthe theory of error-correcting codes: toconstruct codes which detect and correct asmany errors as possible while lengthening themessages the least possible, and which areeasy to decode.

The algebra of finite fields appliesnaturally to codes because they use afinite alphabet

Mathematics has been used in these ques-tions for a long time. Already in 1948, theAmerican mathematician Claude Shannon, oneof the fathers of information theory, obtainedgeneral theoretical results affirming that thereexist codes having optimal qualities in a pre-cise technical sense. However, even thoughShannon's theorem proved the existence ofvery good error-correcting codes, it did notprovide a practical method for constructingthem. In the 1950’s, one only had modest-per-formance error-correcting codes like Hammingcodes, named after their inventor, the Americanmathematician Richard W. Hamming (1915-1998) (in these codes, which were extensivelyused, the control bits are derived from theinformation bits by simple linear equations).

Communicating without errors... 85

The specialists then started studying error-correcting codes and their properties in a sys-tematic manner with the aim of concretelyconstructing codes which would be as effi-cient, or almost as efficient, as the ones pre-dicted by the theoretical results of Shannon.To do this algebra was used extensively. If infor-mation is encoded directly in the binary ̀ àlpha-bet'' of 0 and 1, the underlying algebra is thatof even and odd, known already to Plato (even+ even = even, even + odd = odd, even x even= even, odd x odd = odd, etc.). In fact, it turnsout to be more interesting to consider enco-ding ̀ àlphabets'' having more than two digitsand to translate the result into binarysequences of 0 and 1 only at the end of theprocedure. As an alphabet consists of a finitenumbers of symbols on which we wish to carryout arithmetic operations, the underlying alge-bra is the one studied in the theory of the

finite fields, created by the young Frenchmathematician Évariste Galois at the begin-ning of the 19th century while studying thesolvability of algebraic equations (a finite fieldis a set of elements which are finite in num-ber and which can be added, multiplied anddivided in a manner analogous to ordinarynumbers, the result of the operations remai-ning inside this set. The set consisting of 0 and1, with the arithmetic rules of even and odd,is the finite field of two elements; it is the sim-plest finite field).

Thus, it was using abstract and elaboratealgebra, related to the theory of finite fields,that very effective error-correcting codes werebuilt, adapted to various types of informationtransmission. Two examples among manyothers are provided by the code used forengraving digital audio discs (which makes itpossible to correct up to approximately 4000consecutive erroneous bits, the equivalent ofa scratch of more than 2 millimetres on thedisc!), and the one used by the space probeMariner 9 to send us images of the planetMars.


Olympus Mons, on the planet Mars, is the biggest volcano in the solarsystem: approximately 600 km in diameter and 27 km in height!This image was obtained in 1971-1972 thanks to the space probeMariner 9. The probe sent its information to Earth using an error-correcting code capable of correcting up to 7 erroneous bits out of 32.In each group of 32 bits, 26 bits were control bits and the 6 otherscarried the actual information. Today, one has even more powerfulerror-correcting codes. (Photo NASA/JPL)

Notwithstanding what this French stamp issued in 1984 says, Éva-riste Galois was not a geometrician but an algebraist. He was a pio-neer of group theory and of the theory of finite fields used, in parti-cular, by specialists in error-correcting codes. Provoked to a duel, Galoisdied when he was barely 21 years old.

A new family of codes making use ofthe algebraic geometry of curves

Abstract algebra is not the only instru-ment available to the specialists in error-cor-recting codes. There is also geometry, morespecifically algebraic geometry. The startingpoint of this vast area of modern mathe-matics is the study of geometric objects -curves, surfaces, etc. - defined by algebraicequations. Every high-school student knows,for example, that a parabola can be definedby an algebraic equation of the type y = ax2

+ bx + c, where x and y are the coordinatesof the points on the parabola. One can alsostudy curves defined over finite fields, i.e.,in the algebraic equations which definethem, the quantities x and y are not arbi-trary numbers but only elements of a cer-tain finite field. By using such curves and theassociated algebra of the co-ordinates oftheir points (which are finite in number), anew family of error-correcting codes wasinaugurated approximately twenty yearsago: geometric codes. They have recentlyallowed one to obtain new results aboutbinary codes and to construct codes whichare even more powerful than the the onespredicted by Shannon's work. On the otherhand, the analysis of geometric codes hasled mathematicians to examine more closelythe number of points on an algebraic curvedefined over a finite field. This is a beauti-ful example of the positive feedback that anapplied field can provide to the theoreticaldiscipline which it uses.

Gilles LachaudInstitut de mathématiques de Luminy,

CNRS, Marseille

Communicating without errors... 87

Some references:

• P. Arnoux, « Codage et mathématiques », Lascience au présent (édition EncyclopædiaUniversalis, 1992).

• P. Arnoux, « Minitel, codage et corps finis »,Pour la Science (mars 1988).

• G. Lachaud et S. Vladut, « Les codes correcteursd’erreurs », La Recherche (juillet-août 1995).

• O. Papini, « Disque compact : « la théorie, c’estpratique ! » dans « Secrets de nombres », Hors-série n° 6 de la revue Tangente (1998).

• O. Papini et J. Wolfmann, Algèbre discrète etcodes correcteurs (Springer-Verlag, 1995).

• J. Vélu, Méthodes mathématiques pour l’informa-tique (Dunod, 1995).

• M. Demazure, Cours d’algèbre — primalité,divisibilité, codes (Cassini, 1997)

Jean-Daniel Boissonnat

Reconstructing a surface from the knowledgeof only some of its points: a problem that one comes across often,

be it in geological exploration, in recording archaeological remains,or in medical or industrial imaging.

w hen we probe the ground in someplaces to find out the configuration of variousgeological layers underneath, or when wewant to map the sea-bed, the number of pointswhere measurements are made is necessarilyfinite. The corresponding surfaces need to bereconstructed starting from this limitedamount of data. The situation is similar for allcomputerised imaging systems (scanners,remote-sensors, three-dimensional imaging,etc.) used in medicine, in industry, in archaeo-logy, etc. The starting point is a real object -

which can be a part of the human body, amachine part, an archaeological remain, a geo-logical structure, or something else.Instruments can measure this real object onlyat a certain number of points from which wehave to reconstruct the shape of the objectvirtually. This is the problem of reconstructingsurfaces (Figure 1). It thus consists in using afinite number of points to provide a geome-trical or a computer representation of theobject which will allow us to visualise it on ascreen, to store it in the memory of a compu-

Reconstruction of surfaces

Figure 1. The reconstruction of a surface starting from a sample of its points: this problem arises in various fields.

ter, to easily carry out calculations, even tomodify the object or to give instructions byremote control to get a copy tooled. In short,once the form of a real object is digitally recor-ded with sufficient precision, there are manypossibilities for action and calculation

The economic and industrial stakes invol-ved in the problem of surface reconstruction,and its fundamental character from a scienti-fic point of view, have led to many works devo-ted to it for some twenty years. But it is onlyvery recently that the specialists have forma-lised the problem in mathematical terms,which enabled them to conceive efficient algo-rithms furnishing a faithful reconstruction.The results of this so-called algorithmic geo-metry were transferred very rapidly to indus-try through the creation of young start-ups(such as Raindrop Geomagic in the UnitedStates), or by launching new products by theleaders in computer-assisted design or in medi-cal imaging (Dassault Systèmes, MedicalSiemens).

Voronoï diagrams and Delaunay tri-angulations, two essential geometrictools

For reconstructing a surface from a massof sample points, a large majority of algorithmsuse a central tool in algorithmic geometry:Delaunay triangulation, named after BorisDelone (1890-1980), a Russian mathematicianwhose name was rendered as Delaunay inFrench. A Delaunay triangulation is defined ina natural way starting from what is called aVoronoï diagram, after the name of theUkrainian mathematician Georgi Voronoï(1868-1908). Let us consider a finite set of pointsin space and call it E. The Voronoï diagram of

E is a division of space into convex cells (shownin blue in Figure 2) where each cell consists ofthe points of space closer to a certain point ofE than to any other point of E. Cells - they areconvex polyhedra - are thus defined in a uniquemanner.

Now, let us connect by line segments thepoints of E whose Voronoï cells are adjacent.The set of these segments constitutes theDelaunay triangulation (shown in green inFigure 2) associated to E. These structures canbe defined in spaces of arbitrary dimension;it is the case dimension three - of the usualspace - which is the most interesting one forsurface reconstruction. Voronoï diagrams(Figures 2 and 3) are among the main subjectsof study in algorithmic geometry, and it is inthe 1980’s that their relationship with thetheory of polytopes (analogues of polyhedrain spaces of dimension higher than three) was

Reconstruction of surfaces 89

Figure 2. The Voronoï diagram (in blue) and Delaunay triangula-tion (in green) of a set of points (marked in red). Voronoï diagramsand Delaunay triangulations are fundamental tools in algorithmicgeometry.

established. Their study in the context of sur-face sampling is much more recent.

Why are Voronoï diagrams and Delaunaytriangulations interesting ? If E is a sample ofn points taken on some surface S, one canshow that the corresponding Voronoï diagramand Delaunay triangulation contain a lot ofinformation about this surface. When thesample is sufficiently dense, it can be shownto provide a precise approximation of the sur-face. For example, the vector which joins apoint P of E to the most distant vertex of itsVoronoï cell is a good approximation of thenormal on surface S at point P.

One should ensure that calculationtimes remain reasonable, that thealgorithms are reliable

Several reconstruction algorithms are nowknown capable of constructing a surface S'which correctly approximates the real surfaceS starting from a finite sample of points on S.What's more, the theory of these algorithmsallows one to calculate an upper boundary forthe difference between S' and S', a boundarywhich obviously depends on the sampling den-sity.

As the data sets provided by measuring ins-truments generally contain several hundredsof thousands - even millions - of points, com-binatorial and algorithmic questions play a cri-tical role. It is, for example, important to knowif the quantity of calculations which Delaunaytriangulations require will remain within a rea-sonable limit or not. In the most unfavourablecases, the number T of calculation steps (i.e., inthe final analysis, the computation time) canbe quadratic; in other words, T is, at worst, pro-

portional to the square of the number of samplepoints. It is, however, assumed that this situa-tion does not arise in the case of well-sampledsurfaces. More precise results were very recentlyobtained in the case of polyhedric surfaces S,i.e. surfaces composed only of polygonal faces:for such surfaces and under weak samplingconditions, the size of the calculation for com-puting the triangulation is at worst proportio-nal to the number of sample points. The caseof smooth surfaces is more delicate; it is cur-rently the object of active research.

The theoretical bounds are not all; itremains to know how to efficiently and rapidlycalculate the triangulation from a data set.Many algorithms are known. The more effi-cient ones are called randomised because theycarry out certain random samplings duringtheir execution. The theory of randomisedalgorithms developed very rapidly in the 1990’s

and has led to precise analyses validated byexperiments. In many cases, and the calcula-tion of the Delaunay triangulation is one ofthem, the introduction of an element of ran-domness allows one not to try optimally tosolve the worst case (which is very improbable),and has led to simple and very efficient algo-rithms on the average. One can thus treat


Figure 3. The Voronoï diagram of a set of points on a curve.

Reconstruction of surfaces 91

samples of 100000 points in some ten seconds(Pentium III with 500 MHz).

If it is important to calculate rapidly, to cal-culate reliably is even more important. This isa delicate question because, in general, com-puters only know how to represent numberswith a finite precision (a finite number of deci-mals). Thus it is impossible to give a represen-tation of numbers having infinitely many deci-mals such as π or √2, which would be at thesame time digital and exact. The accumulationof round-off errors can lead to the abnormalbehaviour of the programs. Although thisbehaviour is well known, it is difficult tocontrol, which makes writing and maintainingreliable algorithms very expensive. A signifi-cant part of current research in algorithmicgeometry is related to these questions andcombines the theory of algorithms, formalcomputation (wherein the computer handlessymbols and not explicit numbers) and com-puter arithmetic. It has already led to the deve-lopment of software libraries which have madeprogramming easy, efficient and reliable, suchas the library CGAL (Computational GeometryAlgorithms Library) developed by an interna-tional collaboration of universities and researchorganisations.

Jean-Daniel BoissonnatINRIA (Institut national de recherche en

informatique et en automatique), Sophia-Antipolis

Some references:

• J.-D. Boissonnat et M. Yvinec, Algorithmic geo-metry (Cambridge University Press, 1998).

• J.-D. Boissonnat et F. Cazals, « Smooth surfacereconstruction via natural neighbour interpola-tion of distance functions », dans Proceedings ofthe 16 th Annual ACM Symposium ofComputational Geometry (2000).

• CGAL, The Computational GeometryAlgorithms Library, http://www.cgal.org.

Jean-Pierre Bourguignon

At the end of the 19th century, there were very few ``geometers'',as mathematicians were formerly called.

In one century, their numbers have augmented considerably.Today, they are facing profound changes in their discipline.

during the course of the 20th century, themathematical community has undergone amajor expansion. From a few hundred mem-bers in 1900, it has passed to tens of thousand(probably about 80000) members 100 yearslater. To make an estimate of this kind it isnecessary at first to agree upon a definitionof the term ̀ `mathematician''. We reserve thisterm for those who have reached the educa-tion level equivalent to the doctoral thesis andwhose profession attaches importance tomathematical research or to the assimilationof its results.This choice may be considered to be a bit res-trictive because it leads us, for example, toexclude from our field of vision almost all tea-chers in secondary schools - a category whosenumbers have also increased considerably inall countries of the world during the secondhalf of the 20th century.

This growth is the result of several simulta-neous processes. First of all, there was the rea-lisation, just after the Second World War, of

the importance of the sciences for economicand industrial development. In addition, newgroups of people have been able to enter theprofession. Such is the case for women,although there are great disparities betweendifferent countries. But at the same time, anacademic community bringing together theparticipants in higher education made itsappearance in almost every country. To giveonly one example, the first mathematiciansfrom sub-Saharan Africa worked for their doc-torate in a university in a Western country orin the Soviet Union.

The next generation often pursued their stu-dies in their own country: in the decade 1990-2000, many countries of sub-Saharan Africaset up autonomous establishments of highereducation and gained their independencefrom this point of view. In the coming years,the expansion will continue with probably aconsiderable reinforcement of the mathe-matical communities, in other countries suchas China and India..

Mathematicians in Franceand in the world

A community of researchers and itsnetwork of learned societies

How are mathematical communities orga-nised? The expansion of the internationalmathematical community was accompaniedby an organisation stimulated by learnedsocieties, almost all of which survive thanksto the devotion and the involvement of volun-teers. Mathematical societies are still quitesmall in size, except for the AmericanMathematical Society, which has nearly 15000members and more than 200 employees.

The first stage occurred on the nationallevel, usually at a time when the authoritiesrealised that the development of the sciencescould represent an economic and a militaryasset. Thus the Société mathématique deFrance (SMF), just as the Société française dephysique, was founded in 1872, immediatelyafter the 1870 defeat by Germany and the

consequent reflection on its causes. This nar-row nationalist perspective is fortunately for-gotten

The International Mathematical Union wascreated in 1896. It continues to be small. Itsmain responsibility is to provide the frame-work for organising the International Congressof Mathematicians, a four-yearly event whichhas become an unavoidable rendez-vous forthe mathematical community on a global scale.Its executive committee also undertakes tonominate the commission which awards theFields medals every four years; they representthe most prestigious award in mathematicsas there is no Nobel Prize in this discipline.

The end of the 20th century saw the emer-gence of intermediate structures at the levelof continents. The example was given by ourAfrican colleagues who created the AfricanMathematical Union as early as the 1980’s.Then came the European Mathematical Society

Mathematicians in France and in the world 93

The earth viewed at night. The world distribution of nocturnal lights is not without reminding us of the centres of mathematical activity.However, not all mathematicians work at night! (Photo C. Mayhew and R. Simmon/NASA-GSFC)


(EMS), whose gestation was long - as forthe European Union - and which bringstogether all the national societies formgeographical Europe and from Israel,and the UMALCA, which unites themathematicians from the Caribbean andfrom South America. These new struc-tures were born of the desire to rein-force collaborations on the scale of asub-continent and to have a representativeinterlocutor in the face of the appearance ofa new political level (as in Europe) or to controlthe draining of resources by North America(as was the case for South America) followingthe painful period of military dictatorships

An increasingly broad presence inindustry and the services

Where are mathematicians employed? Thegreat innovation is that, nowadays, mathema-ticians are present in many sectors of industryand the service industry. There is, however, no``mathematical industry'', such as a chemicalindustry or a pharmaceutical industry. Indeed,the jobs entrusted to people with high mathe-matical competence often carry very differentnames, which makes it difficult to count thenumber of `ìndustrial mathematicians''. Arecent estimate leads one to think that there

are nearly 2000 of them employed this way inFrance. This number is to be compared with thenumber of their academic counterparts (mathe-maticians in universities and those working invarious research organisations) which can beestimated much more reliably as approxima-tely 4000. The division of this academic com-munity between public research organisationsand higher education (approximately 10%against 90%) is a little singular: generally, inother scientific disciplines, choices are differentand a much more important proportion devoteall their time to research, without teachingduties. Which sectors are particularly interes-ted in employing mathematicians? Banks andinsurance companies make an increasinglyintensive use of mathematical competence;the products which they sell often rely upona mathematical construction which is at theirvery base. But it is the same with a certainnumber of high-technology companies inwhich the study of complex systems requires

The IHÉS (Institut des hautes études scientifiques), at Bures-sur-Yvette in the suburbs of Paris, and a discussion betweenmathematicians in its buildings. The IHÉS, devoted to funda-mental mathematics and to theoretical physics, is a prestigiousresearch institute. It has only 7 permanent professors but wel-comes some 200 researchers of different nationalities each year,for variable durations. Recently, some of its mathematicians havestarted to concentrate on problems related to molecular biology.(Photo IHÉS and IHÉS-Outsider Agency)

a mathematical approach made possible bythe powerful computational tools providedby the new generation of computers. Thesenew openings are of a kind which might consi-derably change the image of mathematicsamong students. However, they have not yetbeen completely assimilated by French highereducation; the most frequent reason is theexcessive inertia of the education system,which remains centred around training forthe academic professions.

Mathematicians are confronted witha new situation

These new developments have not beenwithout effect on the the way mathematicsis organised, as much in the establishmentsof higher education and research as at thelevel of publications. The resulting situationhas sometimes been presented as a battlebetween ``pure mathematics'' and `àppliedmathematics''. This way of seeing things isunjustified for at least two reasons. On onehand, historical examples abound of situa-tions where new mathematics was createdat the behest of external demand; on theother hand, the new fields to be conqueredcannot be approached by declaring a prioriwhich part of mathematics will be the keyto the solution of the problems which arise.Many surprise-connections can be noted,which proves that the pure/applied dicho-tomy is unproductive in the final analysis. Itwas in this context of internal tension in themathematical community that the Société demathématiques appliquées et industrielles(SMAI) was born in France in 1983. Twentyyears later, the two societies, SMF and SMAI,have found a way of effective co-operationand together carry out programmes of com-mon interest. They count more than 3000

members between them, of which many inthe SMAI are from well outside the acade-mic community The principal innovationcomes from the possibility of studying moreand more complex systems thanks to the useof models of various kinds. Modelling is todaya method to which one often resorts. This newpassion requires a thorough reflection on thefoundations, including philosophical founda-tions, of this approach. One of the capabili-ties, which it is advisable to develop, is theconfrontation of the model with the realitywhich it is supposed to represent. One cannevertheless underline two major tenden-cies which feed on these new contacts bet-ween mathematics and a world which isexternal to it: a renewal of interest in finitestructures (mathematical structures involvingonly a finite number of elements) and theubiquity of stochastic approaches (involvingrandom processes). In the latter field, Francehas taken a remarkable turn, compared withthe situation in other countries with the samelevel of development, except perhaps for theunderrepresentation of statistics and of dataanalysis. On the other hand, the teaching ofdiscrete mathematics, i.e., having to do withfinite structures, remains quite discrete inFrance: very few higher education courseoffer a sufficiently complete training in thisfield Recently, on the occasion of a conferencedevoted to the history of geometry in thesecond half of the 20th century, Stephen Smale,an American mathematician who is one of thefathers of modern topology and who has sincetaken a great interest in numerical analysis,made a pertinent remark: the extraordinarygrowth of mathematics today is often due topeople whom mathematicians tend not toconsider as belonging to their community.

It is true that statistics, cybernetics, ope-rations research, control theory are often


poorly represented in university mathematicsdepartments, whereas the heart of all thesedisciplines is really mathematical. One couldsay the same thing about a good part of theo-retical computer science: the depth and theforce of the organic links it maintains withmathematics are not always appreciated bythe mathematicians themselves. This situationopens up the possibility of considerable growthof the mathematicial community provided thatthey are less prompt at excluding these newactivities from their field. With greater curio-sity and openmindedness there will be grea-ter stimulation and new spheres of activity,for the greater good of the development ofmathematics itself.

The changes in the profession requirenew training programmes

One of the first things to be recognised isrelated to the practice of the mathematician'sprofession required by these new contacts, apractice which cannot limit itself to provingtheorems.

The pressing need is that a sufficient num-ber of mathematicians with very differentbackgrounds get interested in applications.This requires them learning to talk to specia-lists in other disciplines, and listening well.

Already one notes the introduction of spe-cialised training, in financial mathematics forexample, in various higher education struc-tures throughout the world. Other fields, forwhich important openings outside the aca-demic world have appeared, will certainly seethe light of day, on a scale adapted to theseopenings; it is already the case as in regardto the training of actuaries, and it can be fore-seen that mixed training programmes will be

introduced at the interface of mathematicswith biology and medicine, for example

But allowing too-specialised training pro-grammes to proliferate would be an error fortwo reasons: the narrowness of an approachof this kind on the one hand, and, on the otherhand, the risk of schism in the mathematicalcommunity that such a practice would pre-sent. In order that students perceive in a morenatural way the new orientations accessibleto mathematical methods, major extensivemodifications of the teaching curricula willprobably have to be put into place. One mustcreate a fluidity between the academic worldand the world of industry and the serviceindustry; this is a precondition for the gene-ration of good problems, generally dealingwith new fields, so that rather spontaneouslythese problems will be dealt with with thenecessary level of depth.

Jean-Pierre BourguignonCNRS-IHÉS (Institut des hautes études

scientifiques, Bures-sur-Yvette) etÉcole polytechnique, Palaiseau


Some references:

• B. Engquist et W. Schmid (eds.), Mathematicsunlimited — 2001 and beyond (Springer-Verlag,2001).

• C. Casacuberta, R. M. Miró-Roig, J. M. Ortega,et S. Xambó-Descamps (eds.), Mathematicalglimpses into the 21 st century, Round tables held atthe 3rd european congress of mathematics (SocieteCatalana de Matemàtiques, Barcelona, 2001).


Maurice Mashaal

For those who want to do fundamental research in mathematics, longyears of training and an obvious talent are necessary.

Those with the passion have at their disposalseveral study programmes with varied openings.

in the 17th century, a certain magistrate inToulouse by the name of Pierre de Fermat(1601-1665) spent his leisure time in mathe-matical research and carrying out correspon-dence on this subject. Although it was not hisprofession, Fermat made important mathe-matical discoveries. He was, for example a pio-neer in the introduction of algebraic tech-niques in geometry, and his work on the theoryof numbers made him famous - in particularfor a conjecture that he formulated and whichwas proved only in 1994 (it says that the equa-tion xn + yn = zn doesn't have any solutions x,y, z in positive integers as soon as the fixednumber n is greater than or equal to 3). Fermatwas, in fact, one of the most brilliant mathe-maticians of his century.

The time when a gifted autodidact couldmake significant discoveries in his spare timeis over. Admittedly, it still happens that mathe-matical amateurs, who are not professionalmathematicians, discover and prove a newtheorem from time to time. Not only are such

cases rare, but the results obtained by them areabout questions of detail, at the margins of themain currents of evolution in mathematics.

How to becomea mathematician ?

A mathematics course at the university. (Photo Institut de mathé-matique, Université Bordeaux 1)

No, if somebody today wishes to becomea true mover in mathematics, he has to firstundertake long years of study. Approximatelyeight years after the baccalaureat are neces-sary in order to assimilate the knowledge andthe essential techniques which will allow theapprentice-mathematician to acquire his auto-nomy and to start producing original mathe-matical results on his own.

The traditional route: DEUG,licence, maîtrise, DEA and doctoralthesis

Long years of higher studies, agreed, butwhich ones? The traditional route, in France,consists in going through a first university cycleof two years, then a second cycle of two years,and finally a third cycle of approximately fouryears.

The first cycle is devoted to the DEUG(Diplôme d'études universitaires générales).For future mathematicians, it is generally aquestion of the scientific DEUG called``Mathématiques, informatique et applica-tion aux sciences'' (MIAS), in which teachingis centered around mathematics, computerscience and physics; or the one centeredaround mathematics and computer scienceon one hand, economic or the social scienceson the other, called the DEUG``Mathématiques appliquées et sciencessociales'' (MASS).

The first year of the second cycle at the uni-versity is devoted to the licence degree, thesecond year to the maîtrise degree. It can be alicence and a maîtrise in mathematics for thosewho intend to go in for fundamental researchin mathematics, or a second cycle MASS for

those who are interested in applied mathe-matics with economics and the social sciences,or still a maîtrise in engineering mathematics,geared towards industrial applications, withan emphasis on numerical analysis, modelling,data processing, probability and statistics.

The third cycle starts with one year of DEA(Diplôme d’études approfondies), of whichthere is a large variety (in mathematics, thereare nearly fifty different kinds all over France).It might be a fairly general DEA, covering arather broad spectrum of mathematics, or amore specific DEA, such as a DEA in algorith-mics or a DEA in biomathematics. The choiceof the DEA is decisive; it is generally during thisyear that the student comes in contact withresearch-level mathematics, that he will beconfronted with topics of current interest, thathe will have to delve into research articles some-times published very recently.

The DEA largely determines what follows,namely a doctorate which generally takes threeyears to finish. The student decides upon hisresearch topic, finds a thesis supervisor in a uni-versity department and works on the chosentopic in order to obtain original results by him-self and publish them as one or more papersin professional journals. The doctoral degreeis awarded after writing and publicly defen-ding a thesis in front of a jury consisting of spe-cialists.

Magistères and Grandes écoles,springboards towards fundamentalresearch

Licence, maîtrise, DEA, thesis: such is, insummary, the conventional course of studiesfor becoming a research mathematician inFrance; to that are often added one or seve-ral years of post-doctoral research, paid for

How to become a mathematician ? 99

by a fellowship or through a fixed-durationcontract and sometimes carried out abroad,before the young mathematician succeeds infinding a stable job as a researcher or a tea-cher-researcher. This model is roughly the samein most countries. It is the type of route fol-lowed by people like Andrew Wiles, the Britishmathematician who finally solved the famousconjecture of Fermat in 1994.

In fact, the itinerary which we have justdescribed has several alternatives and impor-tant exceptions. First of all Grandes écolessuch as the Écoles normales supérieures andthe École polytechnique have a tendency todrain off the most brilliant students in mathe-matics in France. To prepare for the entranceexamination to these very selective esta-blishments, the candidates do not go throughthe DEUG but study for two (even three) yearsin the ̀ `classes préparatoires'' of a lycée, cha-racterised by intensive preparation and a moreimportant personal effort. After the entranceexaminations, the ``normaliens'' get admit-ted to the second and then the third cycle ofthe universities ; the ``polytechniciens''undergo two years of training at the Écolepolytechnique itself and then join, if they sowish, the university at the level of the DEA.The passage through the Écoles normalessupérieures or the École polytechnique is notobligatory for those who want to becomemathematicians ; however, it should be reco-gnised that a majority of the posts of resear-chers in fundamental mathematics are occu-pied in France by former normaliens orpolytechniciens.

In addition, several universities propose adegree of ``magistère'', excellent training inthree years which combines the licence, themaîtrise and the DEA, and in which the stu-

dents (for the most part normaliens) are selec-ted on merit after their DEUG or the classepréparatoire. Future researchers are well advi-sed to go through the magistère rather thanthe usual courses.

Let us also mention that there are manybridges between engineering schools and theuniversity. Thus, students from the engineeringschools can join the university for a DEA or adoctoral thesis according to their interests andtheir level of competence. Conversely, univer-sity students can join an engineering school oreven a Grande école after their DEUG undercertain conditions.

Engineering studies: shorter, but alsoless geared towards research

Let us say a few words about the engi-neering schools which generally admit theirstudents through a competitive examinationafter the classes préparatoires. Although it isa priori a question of training engineers ratherthan researchers, the quality of mathematicsteaching is often very good there. Some ofthese schools are particularly appropriate forthose who wish to combine mathematics andan engineering or a technology field such asmechanics, acoustics, computer science, etc.There are also more specialised schools, suchas the ENSAE (École nationale de la statistiqueet de l'administration é}conomique) or theENSAI (École nationale de la statistique et del'analyse de l'information) which train statis-ticians and the EURIA which trains actuaries,etc

An engineering training procures a ratherfast entry into professional life, after four orfive years of higher studies. Obviously, the natureof the work done by a mathematician-engineer


working in a company will not be the same asas that of a researcher working in a universitydepartment: it will consist more in applyingmathematics which is already known to concreteproblems rather than in creating new mathe-matics. However, between these two types ofactivity, one can find all the intermediate stagesaccording the the nature of the company, theorganisation or the department, and accordingto the person and his education. For example,an engineer who has been trained in researchthrough a doctoral thesis and who is workingin a large high-tech company may have to carryout fundamental research.

Lastly, one must know that an enginee-ring-type education is also given by universi-ties, through the Instituts universitaires pro-fessionnalisés (IUP) or a professional maîtrisesuch as the MIAGE (maîtrise en méthodesinformatiques appliquées à la gestion des

entreprises) and theMST (maîtrise desciences et tech-niques). As in theengineering schools,these baccalau-reat+4 type coursesare not particularlyor exclusively cen-tred around mathe-matics. But a DESS(diplôme d'étudessupérieures spécia-lisées), a kind of pro-fessional DEA, cansupplement such aneducation and givethe student a moresubstantial mathe-matical training.There is thus a DESS

in ``Calcul scientifique et informatique'',`Ìngénierie mathématique'', ̀ `Mathématiques,informatique et sécurité de l'information'',``Modélisation stochastique et recherche opé-rationnelle'', etc.: the choice is vast!

The job prospects are better whenother disciplines are given a moreimportant place in the curriculum

What are the job prospects for mathe-matics graduates? For those who have goneup to the doctorate or beyond, the naturaloptions are research and higher education:public research organisations like the CNRS,the INRIA, the CEA, the ONERA, etc., but alsobig companies like the RATP or EDF-GDF,recruit researchers and the universities recruitteacher-researchers; in the same way, theGrandes écoles or engineering schools recruit


In mathematics, more than in the other scientific disciplines, the library is an essential tool - for the stu-dents as well as for the researchers. (Photo Institut de mathématique, Université Bordeaux 1)


Interdisciplinarity, a key for the future

Many people are conscious of the need for opening up mathematics to other disciplines. Top mathe-matics turns out to be useful and necessary in increasingly many fields; conversely, the concrete problemswhich arise in these fields can inspire fruitful fundamental research which results in progress in mathe-matics itself. Within teaching and research institutions, there is a political desire to develop interdisci-plinarity but it has been difficult to put it in practice.

One of the main places where action should be taken is higher education. It is true that at the levelof the DEA or the DESS in mathematics, one can notice a certain opening up towards other fields, butthe situation in the second cycle (licence and maîtrise) of the university appears to be more alarming:``mathematics is taught here in an almost completely monolithic fashion; it is necessary to redesign thecurricula, which have changed very little during the last few decades'', says Jean-Pierre Bourguignon,director of the Institut des hautes études scientifiques (IHES). ``For example, the interface betweenmathematics and biology or medicine is almost nonexistent, and it is the same for discrete mathema-tics''. All the same, one can note a few changes, such as the introduction of a paper in modelling in theagrégation examination.

Another place where action should be taken towards greater interdisciplinarity concerns the recruit-ment of researchers and university teachers, as well as, the promotion policy in their careers. As Jean-MarcDeshouillers, director of the Mission scientifiques universitaire at the Ministère de la Recherche under-lines it, `òne can support interdisciplinary exchanges through recruitment committees'', so that, forexample, specialists in statistics are recruited in a biology laboratory. One can also create new laborato-ries devoted to multi-disciplinary topics, or try to modify the focus of the existing laboratories after theirevaluation. This is what organisations like the CNRS or the Ministère de la Recherche are already doing.But, on the way to interdisciplinarity, there are numerous difficulties: it is necessary to give up certainpractices, to circumvent administrative or statutory obstacles, to overcome the incomprehension betweenresearchers of different disciplines, to invest in men and in finances, etc. Things are still at a beginning.``Scientific specialisation and competition, and the system of evaluation and recruitment, very often tendto favour conventional profiles and are not very flexible'', says Christian Peskine, deputy scientific direc-tor for mathematics at the CNRS; the system does not favour the emergence of people having a differentbackground or those wanting to take (scientific) risks in new fields. But those who already have a role ora place in interdisciplinary topics could have an exemplary effect and encourage other colleagues or stu-dents to imitate them.

teachers and also researchers when they haveresearch laboratories. However, only a verylimited number of research and higher-edu-cation jobs are available, and the process is,therefore, very selective. For example, CNRS(Centre national de recherche scientifique)recruits about fifteen young people as ̀ `chargéde recherche'' in mathematics every year (20in 1995, 13 in 1997), the universities about ahundred ̀ `maître de conférence'' (116 in 1995,111 in 1997); these numbers are to be com-pared with the number of doctoral degreesin mathematics awarded every year, which isaround 350-400 (in France).

As for private companies, traditionally theyrecruit engineers; few mathematicians (mea-ning research mathematicians) find a placethere. However, the need for intensive mathe-matical research is being felt in a number emer-ging fields (finance, insurance, informationtechnology, telecommunications, robotics,aeronautics and space industry, oil prospec-ting, etc.). As a result, the presence of mathe-maticians in such companies is going toincrease; and they will be hired all the moreeasily if their mathematical training will havegiven place to other disciplines (see the box).

Mathematical studies at a level less thanthe doctorate offer many more outlets, butthe corresponding jobs are further removedthan that of a mathematician, strictly spea-king. A large number of places is offered bysecondary education: a teacher's job in a lycéecan be had after a licence or a maîtrise, fol-lowed by one year of preparation for the com-petitive examination CAPES (after the licence)or the agrégation (after the maîtrise). Butthere is an array of employment opportuni-ties in banks, in insurance, in information tech-nology, in the ``research and development''

departments of companies, etc., where mathe-matical competence is needed. Provided thatthe studies include a specialisation in one ormore disciplines in addition to mathematics,the risk of finding oneself without work isquite low.

Maurice Mashaaljournaliste scientifique


Some references:

• Infosup n° 189, janvier-février 2001 (Dossierde l’ONISEP sur les études universitaires demathématiques et leurs débouchés).

• Site Internet de l’ONISEP (Office nationald’information sur les enseignements et les pro-fessions) : http://www.onisep.fr.

• Mathématiques à venir - où en est-on à la veillede l’an 2000 ? supplément au n° 75 de laGazette des mathématiciens, publié par la SMF etla SMAI (1997).


la brochure « L’explosion des mathématiques », conçue par la Société mathématique de France (SMF) etla Société de mathématiques appliquées et industrielles (SMAI), a été réalisée avec le soutien financier duMinistère de la Recherche et du CNFM (Comité national français des mathématiciens).

Les éditeurs remercient chaleureusement Madame Brigitte Vogler, chef de la Mission de la Culture etde l’Information scientifiques et techniques et des Musées, au Ministère de la Recherche.

Conception éditoriale et coordination Mireille Martin-Deschamps, Patrick Le Tallec et Michel Waldschmidt,

avec la participation de Fabian Astic, Francine Delmer et Maurice Mashaal.

Comité de lecture Fabian Astic, Jean-Michel Bismut, Jean-Pierre Bourguignon, Mireille Chaleyat-Maurel,

Francine Delmer, Mireille Martin-Deschamps, Patrick Le Tallec,Gérard Tronel, Michel Waldschmidt.

RédactionMaurice Mashaal

Recherche iconographiqueElectron libre, Francine Delmer et Maurice Mashaal

Maquette et mise en pagePatricia Rocher (École polytechnique, Palaiseau)

CouvertureChristophe Bongrain

Réalisation et impressionÉcole polytechnique, Palaiseau

© SMF et SMAI, juillet 2002ISBN 2-85629-120-1

Les titres, intertitres, textes de présentation et légendes ont été établis sous la responsabilité de la rédaction.

SMFInstitut Henri Poincaré

11 rue Pierre et Marie Curie75231 Paris Cedex 05, France

Tel. 01 44 27 67 96http://smf.emath.fr

SMAIInstitut Henri Poincaré

11 rue Pierre et Marie Curie75231 Paris Cedex 05, France

Tel. 01 44 27 66 62http://smai.emath.fr

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