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.