The Cours Peccot is a semester-long mathematics course given at the Collège de France. Each course is given by a mathematician under 30 years of age who has distinguished themselves by their promising work. The course consists of a series of lectures on the current research of the Peccot lecturer. Yves Meyer was chosen as Peccot lecturer for 1968-69 and gave the lecture course Nombres de Pisot et nombres de Salem en analyse harmonique. Exceptionally, the Peccot lecturer is also awarded the Peccot Prize.

The Salem prize is awarded by the School of Mathematics at the Institute for Advanced Study in Princeton to a young mathematician who has been judged to have produced outstanding work in a topic of interest to Raphaël Salem, namely Fourier series or related topics. It was founded by the widow of Raphaël Salem, and first awarded in 1968. The Salem Prize for 1970 was awarded to Yves Meyer of the University of Paris at Orsay for his work on algebraic numbers and harmonic analysis. The jury consisted of Antoni Zygmund, Charles Pisot and Jean-Pierre Kahane.

3.1. The Gauss Prize.

The International Mathematical Union and the Deutsche Mathematiker-Vereinigung awards the Carl Friedrich Gauss Prize for Applications of Mathematics. It is awarded every four years at the International Congress of Mathematicians. The first award was made in 2006 and the second award to Yves Meyer in 2010 at the International Congress of Mathematicians held in August 2010 in Hyderabad, India.

3.2. Chairman of the Carl Friedrich Gauss Prize Committee announces the award.

Wolfgang Dahmen, Chairman of the Carl Friedrich Gauss Prize Committee, announced the winner of the Gauss Prize at the International Congress of Mathematicians 2010 in Hyderabad, India.

Honourable President, dear colleagues,

As the chairman of the Gauss Prize committee, I have the pleasure now to announce the Gauss Prize. Let me briefly introduce the committee. The members were Rolf Jeltsch, Servet Martinez Aguilera, and William R Pulleybank.

A brief word on the Gauss Prize itself. As you know, the name Gauss stands for a unique fusion between fundamental contributions in mathematics in so many areas and concrete applications. The back side of the medal shows one such example, namely, the little circle you see there is the small asteroid Ceres. Gauss had developed a new method to predict its re-appearance, and as a by-product, he developed the least squares method which you could view as the father of all statistical estimators symbolised by the little square in the medal that you see. In that very spirit, the award is for outstanding mathematical contributions with a significant and lasting impact on applications, in particular, outside mathematics.

The person to be awarded has been, in the true sense of the word, in fact, in a double sense in this case, in the centre of some activities nicely indicated by this picture (screen display) from a conference that had taken place in 1992 in Oberwolfach. It is now my great pleasure to reveal the identity of this person: the prize is going to be awarded to Professor Yves Meyer. The brief citation is:-

The IMU and DMV (Deutsche Mathematiker Vereinigung) jointly awarded this prize for his fundamental contributions to those results at the interface between harmonic analysis, number theory and operator theory that finally culminated in the new paradigm referred to as multi-resolution analysis with wavelet bases as the focal point. This paradigm really revolutionised modern methodologies in signal processing but had also strong impact far beyond on other application areas such as non-parametric statistical estimation, and even to pre-conditioning systems of equations that appear in large scale numerical simulation. He really created a new way of multi-resolution thinking which convinced the Gauss committee that Professor Meyer is an outstanding candidate in the very spirit of the award.

3.3. The 2010 award.

Yves Meyer, Professor Emeritus at École Normale Supérieure de Cachan, France, has been selected for the 2010 Gauss Prize:-

... for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis.

3.4. Yves Meyer's Work.

"Whenever you feel competent about a theory, just abandon it." This has been Meyer's principle in his over four decades of outstanding mathematical research work. He believes that only researchers that are new-born to a theme, can show imagination and have big contributions. In this sense, Meyer has had four distinct phases of research activity corresponding to his explorations in four disparate areas - quasicrystals, Calderón-Zygmund programme, wavelets and Navier-Stokes equation. The varied subjects that he has worked on are indicative of his broad interests. In each one of them Meyer has made fundamental contributions. His extensive work in each would suggest that he does not leave a field of research that he has entered until he is convinced that the subject has been brought to its logical end. It is as if Meyer appears on the scene, ties up various loose ends and gives a unifying picture of the existing disparate approaches, which lays the foundation for a proper theoretical framework that has the Meyer stamp on it and he leaves the scene.

The seeds for Meyer's highly original approach in every branch of mathematics that he has ventured into were perhaps sown early in his career. He started his research career after having been a high school teacher for three years following his university education. He completed his Ph.D. in 1966 in just three years. "I was my own supervisor when I wrote my Ph.D.," Meyer has said. This individualistic perspective to a problem has been his hallmark till this day.

In 1970 Meyer introduced some totally new ideas in harmonic analysis (a branch of mathematics that studies the representation of functions or signals as a superposition of some basic waves) that turned out to be not only useful in number theory but also in the theory of the so-called quasicrystals. There are certain algebraic numbers called the Pisot-Vijayaraghavan numbers and certain numbers known as Salem numbers. These have some remarkable properties that show up in harmonic analysis and Diophantine approximation (approximation of real numbers by rational numbers). For instance, the Golden Ratio is such a number. Yves Meyer studied these numbers and proved a remarkable result. Meyer's work in this area led to notions of Meyer and model sets which played an important role in the mathematical theory of quasicrystals.

Quasicrystals are space-filling structures that are ordered but lack translational symmetry and are aperiodic order in general. Classical theory of crystals allows only 2, 3, 4, and 6-fold rotational symmetries, but quasicrystals display 5-fold symmetry and symmetry of other orders. Just like crystals quasicrystals produce modified Bragg diffraction, but where crystals have a simple repeating structure, quasicrystals exhibit more complex structures like aperiodic tilings. Penrose tilings is an example of such an aperiodic structure that displays five-fold symmetry. Meyer studied certain sets in the n-dimensional Euclidean space (now known as a Meyer set) which are characterised by a certain finiteness property of its set of distances. Meyer's idea was that the study of such sets includes the study of possible structures of quasicrystals. This formal basis has now become an important tool in the study of aperiodic structures in general.

In 1975 Meyer collaborated with Ronald Coifman on what are called Calderón- Zygmund operators. The important results that they obtained gave rise to several other works by others, which have led to applications in areas such as complex analysis, partial differential equations, ergodic theory, number theory and geometric measure theory. This approach of Meyer and Coifman can be looked upon as the interplay between two opposing paradigms: the classical complex-analytic approach and the more modern Calderón-Zygmund approach, which relies primarily on real-variable techniques. Nowadays, it is the latter approach that dominates, even for problems that actually belong to the area of complex analysis.

The Calderón-Zygmund approach was the result of the search for new techniques because the complex-analytic methods broke down in higher dimensions. This was done by S Mihlin, Calderón and A Zygmund who investigated and resolved the problem for a wide class of operators, which we now refer to as singular integral operators or Calderón-Zygmund operators. These singular integral operators are much more flexible than the standard representation of an operator, according to Meyer. His collaborative work with Coifman on certain multilinear integral operators has proved to be of great importance to the subject. With Coifman and Alan MacIntosh he proved the boundedness and continuity of the Cauchy integral operator, which is the most famous example of a singular integral operator, on all Lipschitz curves. This had been a long-standing problem in analysis.

Meyer calls the research phase on wavelets, which have had a tremendous impact on signal and image processing, as having given him a second scientific life. A wavelet is a brief wave-like oscillation with amplitude that starts out at zero, increases and decreases back to zero, like what may be recorded by a seismograph or heart monitor. But in mathematics these are specially constructed to satisfy certain mathematical requirements and are used in representing data or other functions. As mathematical tools they are used to extract information from many kinds of data including audio signals and images. Sets of wavelets are generally required to analyse the data. Wavelets can be combined with portions of an unknown signal by the technique of convolution to extract information from the unknown signal.

Representation of functions as a superposition of waves is not new. It has existed since the early 1800s when Joseph Fourier discovered that he could represent other functions by superposing sines and cosines. Sine and cosine functions have well defined frequencies but extend to infinity; that is, while they are localised in frequency, they are not localised in time. This means that although we might be able to determine all the frequencies in a given signal, we do not know when they are present. For this reason a Fourier expansion cannot represent properly transient signals or signals with abrupt changes. For decades scientists have looked for more appropriate functions than these simple sin and cosine functions to approximate choppy signals.

To overcome this problem several solutions have been developed in the past decades to represent a signal in the time and the frequency domain at the same time. The effort in this direction began in the 1930s with the Wigner transform, a construction by Eugene Wigner, the famous mathematician-physicist. Basically wavelets are building blocks of function spaces that are more localised than Fourier series and integrals. The idea behind the joint time-frequency representations is to cut the signal of interest into several parts and analyse each part separately with a resolution matched to its scale. In wavelet analysis, appropriate approximating functions that are contained in finite domains and thus become very suitable for analysing data with sharp discontinuities.

The fundamental question that the wavelet approach tries to answer is how to cut the signal. The time-frequency domain representation itself has a limitation imposed by the Heisenberg uncertainty principle that both the time and frequency domains cannot be localised to arbitrary accuracy simultaneously. Therefore, unfolding a signal in the time-frequency plane is a difficult problem which can be compared with writing the score and listening to the music simultaneously. So groups in diverse fields of research developed techniques to cut up signals localised in time according to resolution scales of their interest. These techniques were the precursors of the wavelet approach.

The wavelet analysis technique begins with choosing a wavelet prototype function, called the mother wavelet. Time resolution analysis can be performed with a contracted, high-frequency version of the mother wavelet. Frequency resolution analysis can be performed with a dilated, low-frequency version of the same wavelet. The wavelet transform or wavelet analysis is the most recent solution to overcome the limitations of the Fourier transform. In wavelet analysis the use of a fully scalable modulated window solves the signal-cutting problem mentioned earlier. The window is shifted along the signal and for every position the spectrum (the transform) is calculated. Then this process is repeated several times with a slightly shorter (or longer) window for every new cycle. The result of this repetitive signal analysis is a collection of time-scale representations of the signal, each with different resolution; in short, multiscale resolution or multiresolution analysis. Simply put the large scale is the big-picture, while the small scale shows the details. It is like zooming in without loss of detail. That is, wavelet analysis sees both the forest and trees.

In geophysics and seismic exploration, one could find models to analyse waveforms propagating underground. Multi-scale decompositions of images were used in computer vision because the scale depended on the depth of a scene. In audio processing, filter banks of constant octave-bandwidth (dilated filters) applied to the analysis of sounds and speech and to handle the problem of Doppler shift multiscale analysis of radar signals were evolved. In physics multiscale decompositions were used in quantum physics by Kenneth G Wilson for the representation of coherent states and also to analyse the fractal properties of turbulence. In neuro-physiology, dilation models had been introduced by the physicist G Zweig to model the responses of simple cells in the visual cortex and in the auditory cochlea. Wavelet analysis would bring these disparate approaches together into a unifying framework. Meyer is widely acknowledged as one of the founders of wavelet theory.

In 1981, Jean Morlet, a geologist working on seismic signals had developed what are known as 'Morlet wavelets', which performed much better than the Fourier transforms. Actually Morlet and Alex Grossman, a physicist whom Morlet had approached to understand the mathematical basis of what he was doing, were the first to coin the term wavelet in 1984. Meyer heard about the work and was the first to realise the connection between Morlet's wavelets and earlier mathematical constructs, such as the work of Littlewood and Paley used for the construction functional spaces and for the analysis of singular operators in the Calderón-Zygmund programme.

Meyer studied whether it was possible to construct an orthonormal basis with wavelets. (An orthonormal basis is like a coordinate system in the space of functions and, like the familiar coordinate axes, each base function is orthogonal to the other. With an orthonormal basis you can represent every function in the space in terms of the basis functions.) This led to his first fundamental result in the subject of wavelets in a Bourbaki seminar article which constructs a whole lot of orthonormal bases with Schwarz class functions (functions which have values only over a small region and decay rapidly outside). This article was a major breakthrough that enabled subsequent analysis by Meyer. "In this article," says Stéphane Mallat, "the construction of Meyer had isolated the key structures in which I could recognise similarities with the tools used in computer vision for multiscale image analysis and in signal processing for filter banks."

A Mallat-Meyer collaboration resulted in the construction of mathematical multiresolution analysis, and a characterisation of wavelet orthonormal bases with conjugate mirror filters that implement a first wavelet transform algorithm that performed faster than the Fast Fourier Transform (FFT) algorithm. Thanks to the Meyer-Mallat result, wavelets became much easier to use. One could now do a wavelet analysis without knowing the formula for the mother wavelet. The process was reduced to simple operations of averaging groups of pixels together and taking differences, over and over. The language of wavelets also became more comfortable to electrical engineers.

3.5. Laudation Abstract.

Yves Meyer has made numerous contributions to mathematics, several of which will be reviewed here, in particular in number theory, harmonic analysis and partial differential equations.

His work in harmonic analysis led him naturally to take an interest in wavelets, when they emerged in the early 1980s; his synthesis of the advanced theoretical results in singular integral operator theory, established by himself and others, and the requirements imposed by practical applications, led to enormous progress for wavelet theory and its applications. Wavelets and wavelet packets are now standard, extremely useful tools in many disciplines; their success is due in large measure to the vision, the insight and the enthusiasm of Yves Meyer.

3.6. Laudation Conclusions.

The scientific life of Yves Meyer combines deep theoretical achievements in harmonic analysis, number theory, partial differential equations and operator theory, with a constant quest for a truly interdisciplinary exchange of ideas and the development of relevant and concrete applications.

This is illustrated most notably by his leading role in the development of wavelet theory, in which his research in harmonic analysis and operator theory led him naturally to the development of the computational multiscale methods that are at the heart of numerous applications of wavelets and wavelet packets in information science and technology.

His pioneering role is clear from the record. But to all his students and collaborators, Yves Meyer also stands out by other characteristics, maybe less tangible in the written record - his insatiable curiosity and drive to understand, his openness to other fields, his boundless enthusiasm and energy that inspired many young scientists, not all of them mathematicians, and the selfless generosity with which he untiringly promoted their work.

3.7. Yves Meyer's CV.

Graduated from École Normale Supérieure, Paris, 1960.

Ph.D., Université de Strasbourg, 1966.

Positions held:

High School teacher, 1960-1963.

Instructor at Université de Strasbourg, 1963-1966.

Professor at Université Paris-Sud, 1966-1980.

Professor at École Polytechnique, 1980-1986.

Professor at Université Paris-Dauphine, 1986-1995.

Research position at Centre National de la Recherche Scientifique (CNRS), 1995-1999.

Professor at École Normale Supérieure de Cachan, 1999-2009.

Professor Emeritus at École Normale Supérieure de Cachan, 2009 -.

4.1. The Abel Prize.

The Abel Prize is awarded by the Norwegian Academy of Science and Letters. The choice of laureate is based on the recommendation of the Abel Committee, which is composed of five internationally recognised mathematicians. The members of the current committee are: John Rognes (chair), Marta Sanz­Solé, Luigi Ambrosio, Marie­France Vignéras and Ben J Green. The Abel Prize and associated events are funded by the Norwegian Government.

4.2. Yves Meyer receives the Abel Prize.

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2017 to Yves Meyer (77) of the École normale supérieure Paris-Saclay, France:-

... for his pivotal role in the development of the mathematical theory of wavelets.

4.3. The Abel Prize Press Release 2017.

Yves Meyer was the visionary leader in the modern development of this theory, at the intersection of mathematics, information technology and computational science.

Wavelet analysis has been applied in a wide variety of arenas as diverse as applied and computational harmonic analysis, data compression, noise reduction, medical imaging, archiving, digital cinema, deconvolution of the Hubble space telescope images, and the recent LIGO detection of gravitational waves created by the collision of two black holes.

The President of the Norwegian Academy of Science and Letters, Ole M Sejersted, announced the winner of the 2017 Abel Prize at the Academy in Oslo today, 21 March.

Yves Meyer will receive the Abel Prize from His Majesty King Harald V at an award ceremony in Oslo on 23 May.

The Abel Prize recognises contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of 6 million NOK (about 675,000 Euro or 715,000 USD).

An intellectual nomad.

Having made important contributions to the field of number theory early in his career, Meyer's boundless energy and curiosity prompted him to work on methods for breaking down complex mathematical objects into simpler wavelike components - a topic called harmonic analysis. This led him in turn to help construct a theory for analysing complicated signals, with important ramifications for computer and information technologies. Then he moved on again to tackle fundamental problems in the mathematics of fluid flow. "During my professional life I obsessively tried to cross the frontiers," he says.

Meyer's work has a relevance extending from theoretical areas of mathematics to the development of practical tools in computer and information science. As such it is a perfect example of the claim that work in pure mathematics often turns out to have important and useful real-world applications.

Yves Meyer has inspired a generation of mathematicians who have gone on to make contributions in their own right. His collaborator on wavelet theory Stéphane Mallat calls him a "visionary" whose work cannot be labelled either pure or applied mathematics, nor computer science either, but simply "amazing".

4.4. The Abel Prize 2017 Citation.

Fourier analysis provides a useful way of decomposing a signal or function into simply-structured pieces such as sine and cosine waves. These pieces have a concentrated frequency spectrum, but are very spread out in space. Wavelet analysis provides a way of cutting up functions into pieces that are localised in both frequency and space. Yves Meyer was the visionary leader in the modern development of this theory, at the intersection of mathematics, information technology and computational science.

The history of wavelets goes back over a hundred years, to an early construction by Alfréd Haar. In the late 1970s the seismologist Jean Morlet analysed reflection data obtained for oil prospecting, and empirically introduced a new class of functions, now called "ondelettes" or "wavelets", obtained by both dilating and translating a fixed function.

In the spring of 1985, Yves Meyer recognised that a recovery formula found by Morlet and Alex Grossmann was an identity previously discovered by Alberto Calderón. At that time, Yves Meyer was already a leading figure in the Calderón-Zygmund theory of singular integral operators. Thus began Meyer's study of wavelets, which in less than ten years would develop into a coherent and widely applicable theory.

The first crucial contribution by Meyer was the construction of a smooth orthonormal wavelet basis. The existence of such a basis had been in doubt. As in Morlet's construction, all of the functions in Meyer's basis arise by translating and dilating a single smooth "mother wavelet", which can be specified quite explicitly. Its construction, though essentially elementary, appears rather miraculous.

Stéphane Mallat and Yves Meyer then systematically developed multiresolution analysis, a flexible and general framework for constructing wavelet bases, which places many of the earlier constructions on a more conceptual footing. Roughly speaking, multiresolution analysis allows one to explicitly construct an orthonormal wavelet basis from any bi-infinite sequence of nested subspaces of $L^{2}(\mathbb{R})$ that satisfy a few additional invariance properties. This work paved the way for the construction by Ingrid Daubechies of orthonormal bases of compactly supported wavelets. In the following decades, wavelet analysis has been applied in a wide variety of arenas as diverse as applied and computational harmonic analysis, data compression, noise reduction, medical imaging, archiving, digital cinema, deconvolution of the Hubble space telescope images, and the recent LIGO detection of gravitational waves created by the collision of two black holes. Yves Meyer has also made fundamental contributions to problems in number theory, harmonic analysis and partial differential equations, on topics such as quasi-crystals, singular integral operators and the Navier-Stokes equations. The crowning achievement of his pre-wavelets work is his proof, with Ronald Coifman and Alan McIntosh, of the $L^{2}$-boundedness of the Cauchy integral on Lipschitz curves, thus resolving the major open question in Calderón's program. The methods developed by Meyer have had a long-lasting impact in both harmonic analysis and partial differential equations. Moreover, it was Meyer's expertise in the mathematics of the Calderón-Zygmund school that opened the way for the development of wavelet theory, providing a remarkably fruitful link between a problem set squarely in pure mathematics and a theory with wide applicability in the real world.

4.5. A biography of Yves Meyer.

Yves Meyer, professor emeritus at the École Normale Supérieure Paris-Saclay in France, proves that, in contrast to what F Scott Fitzgerald said about American lives; in mathematics a life can indeed have a second act, and perhaps even several more. Having made important contributions in the field of number theory early in his career, Meyer's boundless energy and curiosity prompted him to work on methods for breaking down complex mathematical objects into simpler wavelike components - a topic called harmonic analysis. This led him in turn to help construct a theory for analysing complicated signals, with important ramifications for computer and information technologies. Then he moved on again to tackle fundamental problems in the mathematics of fluid flow.

That tendency to cross boundaries was with him from the start. Born on 19 July 1939 of French nationality, he grew up in Tunis on the North African coast. "The Tunis of my childhood was a melting pot where people from all over the Mediterranean had found sanctuary," he said in a 2011 interview. "As a child I was obsessed by the desire of crossing the frontiers between these distinct ethnic groups."

Meyer entered the élite Ecole Normale Supérieure de la rue d'Ulm in Paris in 1957, coming first in the entrance examination. "If you enter ENS-Ulm, you know that you are giving up money and power," he later said. "It is a choice of life. Your life will be devoted to acquiring and transmitting knowledge."

After graduating, Meyer completed his military service as a teacher in a military school. But despite his deep commitment to education and his students, he wasn't suited to the role. "A good teacher needs to be much more methodical and organised than I was," he admits. Moreover, he was uncomfortable with being the one who was "always right". "To do research," Meyer has said, "is to be ignorant most of the time and often to make mistakes." Nevertheless, he feels his experience of high school teaching shaped his life: "I understood that I was more happy to share than to possess."

He joined the University of Strasbourg as a teaching assistant, and in 1966 he was awarded a PhD there - officially under Jean-Pierre Kahane, but Meyer asserts that, like some others in France at that time, he essentially supervised himself. He became a professor of mathematics first at the Université Paris-Sud (as it is now known), then the Ecole Polytechnique and the Université Paris-Dauphine. He moved to the ENS Cachan (recently renamed the ENS Paris-Saclay) in 1995, where he worked at the Centre of Mathematics and its Applications (CMLA) until formally retiring in 2008. But he is still an associate member of the research centre.

Searching for structure

Yves Meyer's work has, in the most general terms, been concerned with understanding mathematical functions with complex and changing forms: a character that can be described by so-called partial differential equations. Fluid flow, for example, is described by a set of such equations called the Navier-Stokes equations, and in the 1990s Meyer helped to elucidate particular solutions to them - a topic that ranks among the biggest challenges in maths.

Meyer's interest in what might be called the structures and regularities of complicated mathematical objects led him in the 1960s to a theory of "model sets": a means of describing arrays of objects that lack the perfect regularity and symmetry of crystal lattices. This work, which arose from number theory, provided the underpinning theory for materials called quasicrystals, first identified in metal alloys in 1982 but prefigured by quasi-regular tiling schemes identified by mathematical physicist Roger Penrose in 1974. The discovery of quasicrystals by materials scientist Dan Shechtman earned him the 2011 Nobel Prize in chemistry. Meyer has sustained his interest in quasicrystals, and together with Basarab Matei in 2010 he helped to elucidate their mathematical structure.

In the 1970s Meyer made profound contributions to the field of harmonic analysis, which seeks to decompose complex functions and signals into components made of simple waves. Along with Ronald Coifman and Alan McIntosh, he solved a long-standing problem in the field in 1982 by proving a theorem about a construction called the Cauchy integral operator. This interest in harmonic decomposition led Meyer into wavelet theory, which enables complex signals to be "atomised" into a kind of mathematical particle called a wavelet.

Wavelet theory began with the work of, among others, physics Nobel laureates Eugene Wigner and Dennis Gabor, geophysicist Jean Morlet, theoretical physicist Alex Grossmann, and mathematician Jan-Olov Strömberg. During a conversation over the photocopier at the Ecole Polytechnique in 1984, Meyer was handed a paper on the subject by Grossmann and Morlet, and was captivated. "I took the first train to Marseilles, where I met Ingrid Daubechies, Alex Grossmann and Jean Morlet", he says. "It was like a fairy tale. I felt I had finally found my home."

Breaking down complexity

From the mid-1980s, in what he called a "second scientific life", Meyer, together with Daubechies and Coifman, brought together earlier work on wavelets into a unified picture. In particular, Meyer showed how to relate Grossmann and Morlet's wavelets to the work of Argentinian mathematician Alberto Calderón, which had supplied the basis for some of Meyer's most significant contributions to harmonic analysis. In 1986 Meyer and Pierre Gilles Lemarié-Rieusset showed that wavelets may form mutually independent sets of mathematical objects called orthogonal bases.

Coifman, Daubechies and Stéphane Mallat went on to develop applications to many problems in signal and image processing. Wavelet theory is now omnipresent in many such technologies. Wavelet analysis of images and sounds allows them to be broken down into mathematical fragments that capture the irregularities of the pattern using smooth, "well-behaved" mathematical functions. This decomposition is important for image compression in computer science, being used for example in the JPEG 2000 format. Wavelets are also useful for characterising objects with very complex shapes, such as so-called multifractals, and Meyer says that they prompted his interest in the Navier-Stokes equations in the mid-1990s.

In the past twenty years Meyer's passion for the structure of oscillating patterns has led him to contribute to the success of the Herschel deep-space telescope mission, and he is working on algorithms to detect cosmic gravitational waves. Meyer's contribution to image processing is also wide-ranging. In 2001 he proposed a mathematical theory to decompose any image into a "cartoon" and a "texture". This "cartoon plus texture" algorithm is now routinely used in criminal investigations to extract digital fingerprints from a complex background.

In such ways, Meyer's work has a relevance extending from theoretical areas of mathematics such as harmonic analysis to the development of practical tools in computer and information science. As such, it is a perfect example of the claim that work in pure mathematics often turns out to have important and useful real-world applications.

An intellectual nomad

Meyer is a member of the French Academy of Science and an honorary member of the American Academy of Arts and Sciences. His previous prizes include the Salem (1970) and Gauss (2010) prizes, the latter awarded jointly by the International Mathematical Union and the German Mathematical Society for advances in mathematics that have had an impact outside the field. The diversity of his work, reflected in its broad range of application, reflects his conviction that intellectual vitality is kept alive by facing fresh challenges. He has been quoted as saying that when you become too much an expert in a field then you should leave it - but he is wary of sounding arrogant here. "I am not smarter than my more stable colleagues," he says simply. "I have always been a nomad." - intellectually and institutionally.

Some feel that Meyer has not yet had the recognition his profound 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 make important contributions in their own right. His collaborator on wavelet theory Stéphane Mallat calls him a "visionary" whose work cannot be labelled either pure or applied mathematics, nor computer science either, but simply "amazing". His students and colleagues speak of his insatiable curiosity, energy, generosity and openness to other fields. "You must dig deeply into your own self in order to do something as difficult as research in mathematics," Meyer claims. "You need to believe that you possess a treasure hidden in the depths of your mind, a treasure which has to be unveiled."

4.6. Terence Tao describes Yves Meyer's work.

In his famous 1960 essay "The unreasonable effectiveness of mathematics in the natural sciences", Eugene Wigner noted the uncanny ability of mathematical notions and discoveries, that were often pursued for no other reason than their intrinsic structure and beauty, to become highly relevant in describing the physical world. The work of the 2017 Abel laureate, Yves Meyer, exemplifies this ability of pure mathematics to cross over into practical real-world applications.

The Fibonacci numbers 1, 1, 2, 3, 5, 8, . . . are a simple example of an object from pure mathematics that appears in surprising ways in nature. The ratios $\large\frac{1}{1}\normalsize , \large\frac{2}{1}\normalsize , \large\frac{3}{2}\normalsize , \large\frac{5}{3}\normalsize , \large\frac{8}{5}\normalsize , ...$ of consecutive Fibonacci numbers converge extremely rapidly to the famous golden ratio $\phi = \Large\frac{1+ √5}{2}\normalsize = 1.61803 ...$. This number is special in many ways. The powers $\phi, \phi^{2}, \phi^{3}, ...$ of the golden ratio lie unexpectedly close to integers: for instance, $\phi^{11} = 199.005 ...$ is unusually close to 199. Meyer's early work focused on a class of numbers (including the golden ratio) with this property, known as Pisot numbers. He discovered that one could use these Pisot numbers to create sets of points (now known as Meyer sets) in a line, a plane, or in higher dimensions that behaved almost, but not quite, like the periodic sets of points one sees in the integers on the real line, or the grid points of a Cartesian plane. A simple example of such a set would be the collection of numbers, such as $\phi + \phi^{3} + \phi^{4}$, that can be formed by adding together distinct powers of the golden ratio $\phi$. Such sets of points are not perfectly periodic, but have a property known as almost periodicity: any pattern that one sees in the set will recur infinitely often, albeit not at perfectly regular intervals. Meyer was motivated to construct these sets to answer purely theoretical questions in the study of Fourier series (superpositions of sinusoidal waves); but a decade after Meyer's work, it was discovered that Meyer sets could be used to help explain the physical properties of quasicrystals - arrangements of molecules that are not periodic in the way that genuine crystals are, but still behave like crystals in many key ways, such as in their diffraction pattern. (The physical discovery of quasicrystals by Dan Schechtman was recognised by the Nobel Prize in Chemistry in 2011.)

One of Yves Meyer's early research interests was the study of singular integral operators - certain integrals arising in such fields as Fourier analysis, complex analysis, and partial differential equations that are only finite due to delicate cancellations and oscillations in the expressions being integrated. One of the basic tools used to analyse these integrals was the Calderón reproducing formula, that allowed one to express an arbitrary function in space as a combination of simpler objects that were localised in space while also being smooth and somewhat oscillatory. Meanwhile, motivated by applications in geophysics, Morlet and Grossmann were also experimenting with analysing time series data (such as seismic data) in both time and frequency simultaneously, by measuring how these data correlated with windowed cosine waves, where the width of the window varied inversely with the frequency of the wave. Meyer realised that the two transforms were essentially identical to each other; this insight then led to the development by Meyer and others of the wavelet transform that allowed one to efficiently and easily decompose any signal into localised oscillatory objects now known as wavelets. This transform captured many of the beneficial features of the more classical Fourier transform (in particular, the ability to separate out the fine-scale aspects of the data from coarse-scale aspects), while suffering fewer of the drawbacks (in particular, information about spatial features of the data, such as edges or spikes, were much more visible using the wavelet transform than with the Fourier transform). This began the "wavelet revolution" of signal processing in the late 1980s and early 1990s, with the wavelet transform now being routinely used in many basic signal processing tasks such as compression (e.g. in the JPEG2000 image compression format) and denoising, as well as more modern applications such as compressed sensing (reconstructing a signal using an unusually small number of measurements).

Meyer's intuition on the interplay between low and high frequency components of functions led to many important theoretical advances in the fields of harmonic analysis and partial differential equations, ranging from the solution of key open problems such as the boundedness of the Cauchy integral operator on Lipschitz curves (solved by Coifman, McIntosh, and Meyer), to the development of new tools such as compensated compactness, paraproducts, and paradifferential calculus that are now indispensable in the understanding of nonlinear effects in partial differential equations, particularly for equations that govern such oscillatory behaviour as the motion of waves in a medium. For instance, the important but still poorly understood phenomenon of turbulence in fluids, in which the velocity field becomes increasingly oscillatory and fine-scaled in behaviour, can be at least partially explained by considering how various wavelet coefficients of the fluid interact with each other, and using the technical tool of paraproducts to measure the strength of such interactions; this has proven to be influential both in the theoretical analysis of the equations of motion of these fluids, as well as in the numerical methods used to simulate these fluids. Meyer's work and insight has not only advanced the pure and applied sides of mathematical analysis, it has also brought them together in a tightly interconnected fashion.

4.7. Yves Meyer makes wavelets with his mathematical theory.

On May 23rd, King Harald V of Norway, will present the Abel Prize to French mathematician Yves Meyer. Each year, the prize is awarded to a laureate for "outstanding work in the field of mathematics". The award is comparable to a Nobel Prize, and is named after the exceptional Norwegian Niels Henrik Abel, who, in a short life from 1802 to 1829, made dramatic advances in mathematics. Meyer was chosen for his development of the mathematical theory of wavelets.

In the early 1980s French engineer Jean Morlet proposed a powerful new way of analysing seismic data to find oil. The oil industry showed no interest, dismissing his proposal with the blinkered response "if it were true, it would be known already". So he published his ideas in a scientific journal. Yves Meyer came across Morlet's article and immediately recognised the potential of the method, which later came to be known as wavelet theory.

About 200 years ago, Joseph Fourier showed how an arbitrary function or signal could be broken down into a sum of simple components called sine waves. As with Morlet's idea, the initial response to Fourier's method was cool, but its enormous utility soon became evident, and Fourier analysis has become perhaps the most powerful and omnipresent technique in applied mathematics.

A difficulty with Fourier's approach is that while sine waves are infinite in extent, many signals are restricted in time. For example, a musical composition is made up by combining short sounds of different frequencies. In the Fourier transform, the sinusoidal components, which extend over the full dimension of time, must cancel exactly throughout the entire infinite period before the music starts and the eternity after it ends. This makes us question whether such sinusoidal functions are really suitable.

Can we find functions that are more in sympathy with the time-bounded nature of the musical composition? This is where wavelets enter the frame. A wavelet, as the name suggests, is a little wave, oscillating but confined to a narrow time interval. We might think of wavelets as mathematical atoms and of functions as molecules. Then wavelet analysis is the partitioning of functions into their atomic components.

The wavelet transform has the beneficial features of the classical Fourier transform, but is free from some of its drawbacks. Wavelets automatically adapt to the different features of a signal, using a small window or width to look at short time high-frequency aspects and a large window for long-lived low frequency components. As Meyer puts it: "You play with the width of the wavelet to catch the rhythm of the signal."

Morlet used wavelets to solve a specific problem, while Meyer developed the idea into a comprehensive mathematical theory. Meyer was born in Paris in 1939 and grew up in Tunis. In 1957 he moved back to Paris to study, and he is currently emeritus professor at the École Normale Supérieure, Paris-Saclay. Following Meyer's work it became clear that a multitude of wavelets, each suited to a particular range of problems, could be constructed.

Digital data is everywhere, and acoustic and video signals are often localised both in time and in frequency range. Wavelets, which are also localised in time and frequency - or space and scale - are exactly what is required for analysis of such data. Wavelets are now crucial to data compression, noise reduction, medical imaging, fingerprint archival, and digital cinema. Wavelets are also useful in solving the partial differential equations that arise in many practical areas such as weather prediction.

Wavelet theory is at the interface of mathematics, information technology and computational science. It is an excellent example of how a purely mathematical theory often becomes important and useful in real-world applications.

5.1. The Princess of Asturias Foundation.

The Princess of Asturias Foundation is a non-profit private institution whose essential aims are to contribute to extolling and promoting those scientific, cultural and humanistic values that form part of the universal heritage of humanity and consolidate the existing links between the Principality of Asturias and the title traditionally held by the heirs to the Crown of Spain.

His Majesty King Felipe VI was Honorary President of the Foundation since its creation in 1980 until his proclamation as King of Spain on 19th June 2014, following which Her Royal Highness Leonor de Borbón y Ortiz, Princess of Asturias, became the Honorary President of this institution which annually convenes the Princess of Asturias Awards.

Aimed at rewarding the scientific, technical, cultural, social and humanitarian work carried out at an international level by individuals, institutions or groups of individuals or institutions, they are granted in eight categories: the Arts, Literature, Social Sciences, Communication and Humanities, Technical and Scientific Research, International Cooperation, Concord and Sports.

The Awards are presented at a formal ceremony held each year at the Campoamor Theatre in Oviedo.

The Awards Ceremony is considered one of the most important cultural events on the international calendar. Throughout their history, the Awards have received recognition from a variety of sources, such as the UNESCO declaration in 2004 acknowledging their extraordinary contribution to the cultural heritage of humanity.

5.2. 2020 Princess of Asturias award for technical & scientific research.

Yves Meyer (French), Ingrid Daubechies (Belgian and American), Terence Tao (Australian and American), and Emmanuel Candès (French) have made immeasurable, ground-breaking contributions to modern theories and techniques of mathematical data and signal processing. These constitute the foundations and backbone of the digital age (by enabling the compression of graphic files with little loss of resolution), of medical imaging and diagnosis (by enabling accurate images to be reconstructed from a small number of data) and of engineering and scientific research (by eliminating interference and background noise). As regards this last point, these techniques serve as the key, for example, to the deconvolution of Hubble Space Telescope images and have been crucial in the detection by LIGO of gravitational waves resulting from the collision of two black holes. The outstanding contributions of these world leaders in mathematics to modern mathematical data and signal processing are essentially based on two different yet complementary tools: wavelets and compressed sensing or matrix completion.

Yves Meyer and Ingrid Daubechies have led the development of modern mathematical wavelet theory, located at the overlap between mathematics, information technology and computer science. After making important contributions to number theory early on in his career, Meyer began working on methods to divide complex mathematical objects into simpler, wave-like components, which is known as harmonic analysis. In 1984, Meyer read the studies that Jean Morlet, Alex Grossmann and Ingrid Daubechies had carried out on wavelets, which sparked his interest in this field. Mathematical wavelet theory enables images and sounds to be decomposed into mathematical fragments, which capture irregularities in the pattern, while at the same time being manageable. This technique underlies data compression and storage and noise suppression. Together with Daubechies, Meyer brought together previous studies and related them to the analytical tools used in harmonic analysis. This discovery later led to Meyer's demonstration that waves can form mutually independent sets of mathematical objects called orthogonal bases. His work inspired Daubechies to construct orthogonal wavelets with compact support, and later biorthogonal wavelets, which revolutionised the field of engineering. Both worked on the development of wavelet packages, which allow improved adaptation to the particularities of a signal or image. They are currently found in numerous technologies, such as digital image compression, and are used in the JPEG 2000 format.

5.3. Minutes of the Jury.

In the course of its online meeting, the Jury for the 2020 Princess of Asturias Award for Technical and Scientific Research, made up of Jesús del Álamo, Juan Luis Arsuaga Ferreras, César Cernuda Rego, Juan Ignacio Cirac Sasturáin, Miguel Delibes de Castro, Elena García Armada, Clara Grima Ruiz, Amador Menéndez Velázquez, Sir Salvador Moncada, Concepción Alicia Monje Micharet, Ginés Morata Pérez, Enrique Moreno González, Lluis Quintana-Murci, Peregrina Quintela Estévez, Manuel Toharia Cortés, María Vallet Regí, chaired by Pedro Miguel Echenique Landiríbar and with Santiago García Granda acting as secretary, has unanimously decided to bestow the 2020 Princess of Asturias Award for Technical and Scientific and Research on Yves Meyer (France), Ingrid Daubechies (Belgium/USA), Terence Tao (Australia/USA) and Emmanuel Candès (France) for their immeasurable, ground-breaking contributions to mathematical theories and techniques for data processing, which have extraordinarily expanded our sensorial capabilities of observation and which constitute the foundations and backbone of the modern digital age.

For their part, Yves Meyer and Ingrid Daubechies have led the development of the modern mathematical theory of wavelets, which are like mathematical heartbeats that enable us to approach Van Gogh and discover his style or to listen to the music enclosed in the apparent noise of the Universe, among many other applications of all kinds. In short, they enable us to visualise what we cannot see and listen to what we cannot hear.

On the other hand, in addition to the undeniable advances in medical imaging and other diagnostic tests derived from the collaboration between Terence Tao and Emmanuel Candès, their contributions to the techniques of compressed sensing enable us to complete electromagnetic signals or reconstruct melodies from which time has stolen notes.

This Award highlights the social contribution of mathematics and its importance as a cross-cutting element in all branches of science.

5.4. Report from the Basque Centre for Applied Mathematics.

Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès have been awarded for their "pioneering and far-reaching contributions to modern theories and techniques of mathematical data and signal processing"

Yesterday we learned that the Princess of Asturias Award for Scientific and Technical Research 2020 has been granted to four exceptional mathematicians, Yves Meyer, Ingrid Daubechies, Terence Tao and Emmanuel Candès, who have in common the work they have done on the so-called "wavelets". According to the jury, the "pioneering and transcendental" contributions to the modern theories and techniques of mathematical data and signal processing developed by these four researchers "highlight the unifying and transversal role of mathematics in different scientific and engineering disciplines, with practical solutions applicable in many fields, and constitute an example of the usefulness of work in pure mathematics".

It was Yves Meyer himself, in the courses he gave at the Universidad Autónoma de Madrid (UAM) when the theory was being developed at the end of the 1980s, who translated this term into Spanish as Ondículas, as they are a mixture of wave and particle. The theory of wavelets can be understood as the modern version of Fourier Analysis, a mathematical technique that has proved to be very versatile in its applications to both real life and mathematics itself. J-B J Fourier was a French mathematician and physicist of the first half of the nineteenth century.

The use of wavelets has led to unsuspected applications in a wide variety of scenarios, as diverse as computational harmonic analysis, data compression, noise reduction, medical imaging, deconvolution of Hubble Space Telescope images, and the recent detection (LIGO) of gravitational waves created by the collision of two black holes. In essence, it is a matter of breaking down any signal, whether acoustic, optical, or of any kind, or in general any function (for example, series of economic, social data, or image pixels) into its elementary pieces in such a way that knowing only a part you understand the whole, in the same way that from a few musical notes a melody is constructed.

At the Basque Centre for Applied Mathematics - BCAM we have a strong relationship with two of the award winners; Yves Meyer and Terence (Terry) Tao. My relationship with Yves dates back precisely to when he was a PhD student at UAM. As for Terry, the work I did with him, and with UAM professor Ana Vargas in the late 1990s, is one that I am particularly happy about. On the one hand we established a new "record" in a difficult problem about the interaction of waves, a problem that is at the heart of the wave-particle duality and that remains unsolved. And on the other hand, it was an exciting moment in which we saw the work of several years bear fruit; it was undoubtedly a real pleasure to work with both.