The Abel Prize Laureate 2017 - Yves Meyer www.abelprize.no
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The Abel Prize Laureate 2017 Yves Meyer École normale supérieure Paris-Saclay, France www.abelprize.no
Yves Meyer receives the Abel Prize for 2017 “ for his pivotal role in the development of the mathematical theory of wavelets.”
Citation
The Abel Committee
The Norwegian Academy of Science and or “wavelets”, obtained by both dilating infinite sequence of nested subspaces Meyer’s expertise in the mathematics
Letters has decided to award the Abel and translating a fixed function. of L2(R) that satisfy a few additional of the Calderón-Zygmund school that
Prize for 2017 to In the spring of 1985, Yves Meyer invariance properties. This work paved opened the way for the development of
recognised that a recovery formula the way for the construction by Ingrid wavelet theory, providing a remarkably
Yves Meyer, École normale supérieure found by Morlet and Alex Grossmann Daubechies of orthonormal bases of fruitful link between a problem set
Paris-Saclay, France was an identity previously discovered compactly supported wavelets. squarely in pure mathematics and a theory
by Alberto Calderón. At that time, Yves In the following decades, wavelet with wide applicability in the real world.
“for his pivotal role in the Meyer was already a leading figure analysis has been applied in a wide
development of the mathematical in the Calderón-Zygmund theory of variety of arenas as diverse as applied
theory of wavelets.” singular integral operators. Thus began and computational harmonic analysis,
Meyer’s study of wavelets, which in less data compression, noise reduction,
Fourier analysis provides a useful way than ten years would develop into a medical imaging, archiving, digital cinema,
of decomposing a signal or function into coherent and widely applicable theory. deconvolution of the Hubble space
simply-structured pieces such as sine The first crucial contribution by telescope images, and the recent LIGO
and cosine waves. These pieces have a Meyer was the construction of a smooth detection of gravitational waves created
concentrated frequency spectrum, but orthonormal wavelet basis. The existence by the collision of two black holes.
are very spread out in space. Wavelet of such a basis had been in doubt. As in Yves Meyer has also made
analysis provides a way of cutting up Morlet’s construction, all of the functions fundamental contributions to problems
functions into pieces that are localised in Meyer’s basis arise by translating and in number theory, harmonic analysis
in both frequency and space. Yves dilating a single smooth “mother wavelet”, and partial differential equations, on
Meyer was the visionary leader in the which can be specified quite explicitly. topics such as quasi-crystals, singular
modern development of this theory, at the Its construction, though essentially integral operators and the Navier-Stokes
intersection of mathematics, information elementary, appears rather miraculous. equations. The crowning achievement
technology and computational science. Stéphane Mallat and Yves Meyer then of his pre-wavelets work is his proof,
The history of wavelets goes back systematically developed multiresolution with Ronald Coifman and Alan McIntosh,
over a hundred years, to an early analysis, a flexible and general framework of the L2-boundedness of the Cauchy
construction by Alfréd Haar. In the for constructing wavelet bases, which integral on Lipschitz curves, thus resolving
late 1970s the seismologist Jean places many of the earlier constructions the major open question in Calderón’s
Morlet analysed reflection data on a more conceptual footing. Roughly program. The methods developed by
obtained for oil prospecting, and speaking, multiresolution analysis Meyer have had a long-lasting impact
empirically introduced a new class allows one to explicitly construct an in both harmonic analysis and partial
of functions, now called “ondelettes” orthonormal wavelet basis from any bi- differential equations. Moreover, it was
4 5
A biography
of Yves Meyer
Philip de Greff Ball
Yves Meyer, professor emeritus at the in Paris in 1957, coming first in the
École normale supérieure Paris-Saclay in entrance examination. “If you enter ENS-
France, proves that, in contrast to what Ulm, you know that you are giving up
F. Scott Fitzgerald said about American money and power,” he later said. “It is a
lives; in mathematics a life can indeed choice of life. Your life will be devoted to
have a second act, and perhaps even acquiring and transmitting knowledge.”
several more. Having made important After graduating, Meyer completed his
contributions in the field of number theory military service as a teacher in a military
early in his career, Meyer’s boundless school. But despite his deep commitment
energy and curiosity prompted him to to education and his students, he wasn’t
work on methods for breaking down suited to the role. “A good teacher
complex mathematical objects into needs to be much more methodical
simpler wavelike components – a topic and organised than I was,” he admits.
called harmonic analysis. This led him Moreover, he was uncomfortable with
in turn to help construct a theory for being the one who was “always right”.
analysing complicated signals, with “To do research,” Meyer has said, “is to
important ramifications for computer be ignorant most of the time and often to
and information technologies. Then he make mistakes.” Nevertheless, he feels
moved on again to tackle fundamental his experience of high school teaching
problems in the mathematics of fluid flow. shaped his life: “I understood that I was
That tendency to cross boundaries more happy to share than to possess.”
was with him from the start. Born on He joined the University of Strasbourg
19 July 1939 of French nationality, he as a teaching assistant, and in 1966 he
grew up in Tunis on the North African was awarded a PhD there – officially under
coast. “The Tunis of my childhood Jean-Pierre Kahane, but Meyer asserts
was a melting pot where people that, like some others in France at that
from all over the Mediterranean had time, he essentially supervised himself. He
found sanctuary,” he said in a 2011 became a professor of mathematics first
interview. “As a child I was obsessed at the Université Paris-Sud (as it is now
by the desire of crossing the frontiers known), then the École Polytechnique and
between these distinct ethnic groups.” the Université Paris-Dauphine. He moved
Meyer entered the élite École to the ENS Cachan (recently renamed
normale supérieure de la rue d’Ulm the ENS Paris-Saclay) in 1995, where
© B. Eymann – Académie des sciences
he worked at the Centre of Mathematics quasicrystals by materials scientist Dan Daubechies, Alex Grossmann and Jean for image compression in computer
and its Applications (CMLA) until formally Shechtman earned him the 2011 Nobel Morlet”, he says. “It was like a fairy tale. science, being used for example in the
retiring in 2008. But he is still an associate Prize in chemistry. Meyer has sustained I felt I had finally found my home.” JPEG 2000 format. Wavelets are also
member of the research centre. his interest in quasicrystals, and together useful for characterising objects with
with Basarab Matei in 2010 he helped to Breaking down complexity very complex shapes, such as so-called
Searching for structure elucidate their mathematical structure. From the mid-1980s, in what he multifractals, and Meyer says that they
Yves Meyer’s work has, in the most In the 1970s Meyer made profound called a “second scientific life”, prompted his interest in the Navier-
general terms, been concerned with contributions to the field of harmonic Meyer, together with Daubechies and Stokes equations in the mid-1990s.
understanding mathematical functions analysis, which seeks to decompose Coifman, brought together earlier work In the past twenty years Meyer’s
with complex and changing forms: a complex functions and signals into on wavelets into a unified picture. In passion for the structure of oscillating
character that can be described by so- components made of simple waves. particular, Meyer showed how to relate patterns has led him to contribute to the
called partial differential equations. Fluid Along with Ronald Coifman and Alan Grossmann and Morlet’s wavelets to success of the Herschel deep-space
flow, for example, is described by a set McIntosh, he solved a long-standing the work of Argentinian mathematician telescope mission, and he is working on
of such equations called the Navier- problem in the field in 1982 by proving a Alberto Calderón, which had supplied algorithms to detect cosmic gravitational
Stokes equations, and in the 1990s theorem about a construction called the the basis for some of Meyer’s most waves. Meyer’s contribution to image
Meyer helped to elucidate particular Cauchy integral operator. This interest in significant contributions to harmonic processing is also wide-ranging. In 2001
solutions to them – a topic that ranks harmonic decomposition led Meyer into analysis. In 1986 Meyer and Pierre he proposed a mathematical theory to
among the biggest challenges in maths. wavelet theory, which enables complex Gilles Lemarié-Rieusset showed decompose any image into a “cartoon”
Meyer’s interest in what might be signals to be “atomised” into a kind of that wavelets may form mutually and a “texture”. This “cartoon plus
called the structures and regularities of mathematical particle called a wavelet. independent sets of mathematical texture” algorithm is now routinely used
complicated mathematical objects led Wavelet theory began with the objects called orthogonal bases. in criminal investigations to extract digital
him in the 1960s to a theory of “model work of, among others, physics Nobel Coifman, Daubechies and Stéphane fingerprints from a complex background.
sets”: a means of describing arrays of laureates Eugene Wigner and Dennis Mallat went on to develop applications In such ways, Meyer’s work has a
objects that lack the perfect regularity Gabor, geophysicist Jean Morlet, to many problems in signal and image relevance extending from theoretical areas
and symmetry of crystal lattices. This theoretical physicist Alex Grossmann, processing. Wavelet theory is now of mathematics such as harmonic analysis
work, which arose from number theory, and mathematician Jan-Olov Strömberg. omnipresent in many such technologies. to the development of practical tools
provided the underpinning theory for During a conversation over the photo Wavelet analysis of images and sounds in computer and information science.
materials called quasicrystals, first copier at the École Polytechnique in allows them to be broken down into As such, it is a perfect example of the
identified in metal alloys in 1982 but 1984, Meyer was handed a paper on mathematical fragments that capture claim that work in pure mathematics
prefigured by quasi-regular tiling schemes the subject by Grossmann and Morlet, the irregularities of the pattern using often turns out to have important
identified by mathematical physicist and was captivated. “I took the first smooth, “well-behaved” mathematical and useful real-world applications.
Roger Penrose in 1974. The discovery of train to Marseilles, where I met Ingrid functions. This decomposition is important
8 9
An intellectual nomad make important contributions in their own
Meyer is a member of the French right. His collaborator on wavelet theory
Academy of Science and an honorary Stéphane Mallat calls him a “visionary”
member of the American Academy of whose work cannot be labelled either pure
Arts and Sciences. His previous prizes or applied mathematics, nor computer
include the Salem (1970) and Gauss science either, but simply “amazing”. His
(2010) prizes, the latter awarded jointly students and colleagues speak of his
by the International Mathematical Union insatiable curiosity, energy, generosity
and the German Mathematical Society and openness to other fields. “You must
for advances in mathematics that have dig deeply into your own self in order to
had an impact outside the field. The do something as difficult as research in
diversity of his work, reflected in its mathematics,” Meyer claims. “You need
broad range of application, reflects his to believe that you possess a treasure
conviction that intellectual vitality is kept hidden in the depths of your mind, a
alive by facing fresh challenges. He has treasure which has to be unveiled.”
been quoted as saying that when you
become too much an expert in a field Sources
then you should leave it – but he is wary U. Persson, EMS Newsletter, June 2011, 25-28
of sounding arrogant here. “I am not http://mathineurope.eu/images/profession_pdf/
smarter than my more stable colleagues,” newsletter_pdf/meyer_nl80.pdf
he says simply. “I have always been a
nomad – intellectually and institutionally.” Yves Meyer, National Academy of Sciences,
US:
Some feel that Meyer has not
http://www.nasonline.org/member-directory/
yet had the recognition his profound
members/20033195.html
achievements warrant, perhaps because
he has been so selfless in promoting
the careers of others and in devoting
himself to mathematical education
as well as research. “The progress of
mathematics is a collective enterprise,”
he has said. “All of us are needed.”
He has inspired a generation of
mathematicians who have gone on to
10 11A glimpse of the Laureate’s work
If it were true,
it would be known
Arne B. Sletsjøe
When Jean Morlet, a French engineer, It was true, it was known, but until then
in the early 80´s presented his vision of nobody had seen the connection. Again
a new and revolutionary way to collect the beauty of mathematics revealed
seismic data, the oil company showed itself as a new and unexpected link
no faith in his idea and turned him down between different fields appeared.
with the argument: “If it were true, it There is a famous anecdote about the
would be known.” Rather than using the 14-year-old Wolfgang Amadeus Mozart,
wavelet method for finding oil and making visiting the Wednesday service at the
money for the company, the results Vatican. During the service Wolfgang
were published in a scientific journal in listened to Gregorio Allegri´s Miserere,
the spring of 1984. Later that year Yves a beautiful nine-voice setting of Psalm
Meyer was waiting for his turn at the 51 to be performed only during the Holy
photocopier at the École Polytechnique Week. To prevent uncontrolled distribution
in Paris. A colleague was copying an of the piece, the Pope had for more
article about wavelets, written in Marseille than a century forbidden anyone from
by Jean Morlet and the physicist Alex transcribing it. But a couple of hours after
Grossmann. Meyer was handed a copy the service the young Wolfgang were
of the article and realized the similarity able to transcribe the entire piece from
with a mathematical theory he himself memory. A few misprints were corrected
had been studying extensively. Excited after a new visit at the Friday service.
by the unexpected and promising link The underlying principle of composing
between theories of different nature he and performing music has close
took the train to Marseille to join the “Club connections to mathematical theories
of waveletters”. As the oil company had such as Fourier or wavelet analysis.
argued against the new theory for the When composing music, the composer
reason that if it were true, it would be creates melodies and harmonies in his or
known, Meyer approved by his trip to her mind, and then encodes the sounds
Marseille the theory for the same reason. into notes on a notepaper. A mighty final
12 13chord of a Beethoven symphony filling actually written during the Baroque era.
the whole concert hall with sound and The sound of the echoes from layers of
creating goose bumps in the audience oil has not quite the same harmonies as 1
is encoded in a small number of notes in the masterpiece of Allegri, and not even
the conductor’s score. Mozart was able Mozart would have been able to memorize 0,5
to, just by memory, to make the “note ten minutes of seismic raw data, and by
transform” of the Miserere, ensuring that no means been able to interpret them 0
the singers and musicians to this day as information about where to find oil.
have had the pleasure of performing Jean Morlet analysed reflection data -0,5
the “invers note transform”, i.e. create obtained for oil prospecting. A vibration is
the beautiful music from the score. sent into the ground and the echoes are
Some years after Mozart´s visit to collected. The idea is closely related to -6 -4 -2 0 2 4 6
Rome, a French mathematician, Joseph the principle used by the sonar of the bat. Meyer wavelet
Fourier invented a mathematical counter- The problem is to analyse the reflected
part of the “note transform”. The idea signals to extract the valuable information of the wavelet itself, is the “amplitude” The first crucial contribution
of Fourier´s theory is that any stationary about the layers of oil. Jean Morlet for the specific version of the wavelet. by Meyer was the construction of a
signal, e.g. a lasting note from a violin, is analysed these signals, and empirically Since the wavelets are “orthogonal” smooth orthonormal wavelet basis.
built up of pure sinusoidal sound waves of introduced a new class of functions, this decomposition will be unique, and As in Morlet’s construction, all of the
certain frequencies and amplitudes. The which he called “wavelets of constant the inverse process will completely functions in Meyer’s basis arise by
Fourier transform, analogous to the note shape”, after some time shortened to reproduce the original signal. Notice the translating and dilating a single smooth
transform, picks out each frequency and just “wavelets” by the waveletters. similarity with the idea of note transform mother wavelet, which can be specified
the corresponding amplitude. Ignoring in composing and performing music. quite explicitly. Its construction,
the frequencies of neglectable amplitude, Meyer wavelet The Abel Prize for 2017 is awarded though essentially elementary, appears
the original signal can be encoded using The wavelet technique can be described to Yves Meyer for his pivotal role in the rather miraculously. Later on Yves
a few pairs of real numbers. And the nice as follows: The base of the theory is development of the mathematical theory Meyer together with Stéphane Mallat
thing is that by using the invers Fourier the “mother wavelet”, a small part of an of wavelets. Yves Meyer was the visionary systematically developed the theory
transform it is possible to reproduce the oscillating function. The frequency of leader in the modern development of this of multiresolution analysis, a general
original signal with great accuracy. the oscillation varies; the same does the theory, at the intersection of mathematics, framework for constructing wavelet bases.
Fourier analysis is well suited for width of the wavelet. But there is a close information technology and computational The science history takes many
studying stationary signals, whose connection between the two; the higher science. More than a hundred years ago different turns. If Morlet´s wavelet idea
statistical properties do not change the frequency, the smaller the width. The Alfréd Haar constructed an early version was accepted by the oil company, if
over time. In real life only few signals key observation is that the shape of the of a wavelet. Haar´s wavelets had some École Polytechnique in Paris had invested
are stationary, which causes the need wavelet is constant, up to scaling in some nice properties, but unfortunately also in an additional photocopier, if Yves
of a method to efficiently handle signals direction. The principle remains the same some missing aspects. During the Meyer missed the train to Marseille, ...
where sudden and unexpected changes whether the signal is a sound signal or twentieth century many other wavelets
interrupt more evenly distributed signals. reflects the content of an image. The were constructed. Even if the suggested Epilogue. The equivalent statement of the
Allegri´s Miserere is perhaps information of the signal is compared to functions constantly improved the heading; If it is not known, it is not true,
well suited for Fourier analysis, as a differently scaled versions of the mother techniques and gave successful would have led to the end of all science.
masterpiece of often-recorded examples wavelet, and the only data we need to applications, no real breakthroughs
of late Renaissance music, although it was store in addition to the characteristics were achieved.
14 1516 17
About the
Abel Prize
The Abel Prize is an international award Mathematical Union’s Commission for
for outstanding scientific work in the field Developing Countries, and the Bernt
of mathematics, including mathematical Michael Holmboe Memorial Prize for
aspects of computer science, excellence in teaching mathematics in
mathematical physics, probability, Norway. In addition, national mathematical
numerical analysis, scientific computing, contests, and various other projects
statistics, and also applications of and activities are supported in order to
mathematics in the sciences. stimulate interest in mathematics among
The Abel Prize has been awarded children and youth.
since 2003 by the Norwegian
Academy of Science and Letters. —
The choice of laureates is based on
the recommendations from the Abel Call for nominations 2018
Committee. The prize carries a cash The Norwegian Academy of Science and
award of 6 million NOK (about 675,000 Letters hereby calls for nominations for
Euro or about 715,000 USD). the Abel Prize 2018, and invite you (or
The prize is named after the your society or institution) to nominate
exceptional Norwegian mathematician candidate(s). Nominations are confidential
Niels Henrik Abel (1802–1829). According and a nomination should not be made
to the statutes of the Abel Prize the known to the nominee.
objective is both to award the annual Abel
Prize, and to contribute towards raising Deadline for nominations for the Abel Prize
the status of mathematics in society and 2018 is September 15, 2017.
stimulating the interest of children and
young people in mathematics. Please consult www.abelprize.no for
Among initiatives supported are more information.
the Abel Symposium, the International
18The Abel Prize Laureates
2016 Sir Andrew J. Wiles
“for his stunning proof of Fermat’s Last
Theorem by way of the modularity conjecture
for semistable elliptic curves, opening a new
era in number theory.”2015 John Forbes Nash, Jr. 2008 John Griggs Thompson
and Louis Nirenberg and Jacques Tits
“for striking and seminal contributions to the theory of “for their profound achievements in algebra and in
nonlinear partial differential equations and its applications particular for shaping modern group theory.”
to geometric analysis.”
2007 Srinivasa S. R. Varadhan
2014 Yakov G. Sinai “for his fundamental contributions to probability theory
“for his fundamental contributions to dynamical systems, and in particular for creating a unified theory of large
ergodic theory, and mathematical physics.” deviations.”
2013 Pierre Deligne 2006 Lennart Carleson
“for seminal contributions to algebraic geometry and “for his profound and seminal contributions to harmonic
for their transformative impact on number theory, analysis and the theory of smooth dynamical systems.”
representation theory, and related fields.”
2005 Peter D. Lax
2012 Endre Szemerédi “for his groundbreaking contributions to the theory and
“for his fundamental contributions to discrete application of partial differential equations and to the
mathematics and theoretical computer science, and computation of their solutions.”
in recognition of the profound and lasting impact of
these contributions on additive number theory and
ergodic theory.”
2004 Sir Michael Francis Atiyah
and Isadore M. Singer
2011 John Milnor “for their discovery and proof of the index theorem,
“for pioneering discoveries in topology, geometry bringing together topology, geometry and analysis, and
and algebra.” their outstanding role in building new bridges between
mathematics and theoretical physics.”
2010 John Torrence Tate 2003 Jean-Pierre Serre
“for his vast and lasting impact on the theory of numbers.”
“for playing a key role in shaping the modern form of many
parts of mathematics, including topology, algebraic
geometry and number theory.”
2009 Mikhail Leonidovich Gromov
“for his revolutionary contributions to geometry.”
22 23Programme Abel Week 2017
May 22 Abel Banquet at Akershus Castle
— The Norwegian government hosts a
Front page photo: © B. Eymann – Académie des sciences
Holmboe Prize Award Ceremony banquet in honour of the Abel Laureate
The Minister of Education and Research at Akershus Castle (by invitation only)
Torbjørn Røe Isaksen presents the
Bernt Michael Holmboe Memorial Prize
for teachers of mathematics at May 24
Oslo Cathedral School —
— The Abel Lectures
Wreath-laying at the Abel Monument Laureate Lecture, Science Lecture, and
by the Abel Prize Laureate in the other lectures in the field of the Laureate’s
Palace Park work at Georg Sverdrups Hus, Aud. 1,
University of Oslo
—
May 23 The Abel Party
— at The Norwegian Academy of Science
Abel Prize Award Ceremony and Letters
His Majesty King Harald V presents the —
Abel Prize to the Laureate in the University Abel event for school children at the
Aula, Oslo, Norway University of Oslo, Blindern Campus
— 400 school children in the 5th grade
Reception and interview will be invited to take part in different
with the Abel Laureate mathematical games and activities at
TV presenter and journalist Nadia a math circus.
Hasnaoui interviews the Abel Laureate
at Det Norske Teatret
Press contact:
Anne-Marie Astad
anne.marie.astad@dnva.no
+ 47 22 84 15 12 direct line
+ 47 415 67 406 mobile
For other information:
Håkon Sandbakken
The Norwegian Academy
hakon.sandbakken@dnva.no
of Science and Letters
+ 47 454 03 077 mobile
Register online at: www.abelprize.no from mid-April,
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