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Home , Jacques Tits, Lennart Carleson, Pierre Deligne, Ronald Coifman, Yves Meyer

The Laureate 2017

Yves Meyer École normale supérieure -Saclay, www.abelprize.no receives the Abel Prize for 2017 “for his pivotal role in the development of the mathematical theory of .” 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 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 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 , Meyer’s study of wavelets, which in less data compression, noise reduction, 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 , 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 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 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 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 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, Some feel that Meyer has not US: 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 who have gone on to

10 11 A 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 13 chord 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 15 16 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

18 and

The 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 and “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 theory.” to geometric analysis.”

2014 Yakov G. Sinai 2007 Srinivasa S. R. Varadhan “for his fundamental contributions to “for his fundamental contributions to dynamical systems, and in particular for creating a unified theory of large ergodic theory, and mathematical physics.” deviations.”

2013 2006 “for seminal contributions to and “for his profound and seminal contributions to harmonic for their transformative impact on number theory, analysis and the theory of smooth dynamical systems.” , and related fields.”

2012 Endre Szemerédi 2005 Peter D. Lax “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 2011 and Isadore M. Singer “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 outstand ­ing role in 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 2009 Mikhail Leonidovich Gromov geometry and number theory.” “for his revolutionary contributions to geometry.”

22 23 Programme Abel Week 2017

May 22 Abel Banquet at Akershus Castle

— The Norwegian government hosts a Front page photo: 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 © B. Eymann – Académie des sciences 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 [email protected] + 47 22 84 15 12 direct line + 47 415 67 406 mobile

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