# The Abel lectures 2017

Abel Laureate Yves Meyer gave his prize lecture at the University of Oslo on the 24th of May, with following Abel lectures by Stéphane Mallat, Ingrid Daubechies og Emmanuel Jean Candès. Watch the lectures here.

**Welcome **Vice-Rector of the University of Oslo,

**Knut Fægri**,

President of The Norwegian Academy of Science and Letters,

**Ole M. Sejersted**, and Chair of the Abel Committee,

**John Rognes**

**Abel laureate Yves Meyer***,* École Normale Supérieure Paris-Saclay:

*Detection of gravitational waves and time-frequency wavelets*

*,*

Abstract:

Sergey Klimenko designed the algorithm used to detect gravitational waves. This algorithm depends on the time-frequency wavelets which have been elaborated by Ingrid Daubechies, Stéphane Jaffard, and Jean-Lin Journé. After describing the now famous discovery of gravitational waves the focus will be on time-frequency analysis.

### 11:05 - 12:00

**Professor Stéphane Mallat, **École Normale Supérieure:

*A Wavelet Zoom to Analyze a Multiscale World *

Abstract:

Complex physical phenomena, signals and images involve structures of very different scales. A wavelet transform operates as a zoom, which simplifies the analysis by separating local variations at different scales. Yves Meyer found wavelet orthonormal bases having better properties than Fourier bases to characterize local properties of functions, physical measurements and signals. This discovery created a major scientific catalysis, which regrouped physicists, engineers and mathematicians, leading to a coherent theory of multiscale wavelet decompositions with a multitude of applications.* *

This lecture will explain the construction of Meyer wavelet bases and their generalization with fast computations. We shall follow the path of this human adventure, with ideas independently developed by scientists working in quantum physics, geophysics, image and signal processing but also neurophysiology of perception. The synthesis in the 1980's provoked by Yves Meyer's work was an encounter between applications and a pure harmonic analysis research program, initiated by Littlewood-Paley in the 1930's. It remains at the roots of open mathematical problems in high-dimension, for physics and big data analysis.

**Professor Ingrid Daubechies****, **Duke University:

Wavelet bases: roots, surprises and applications

Wavelet bases: roots, surprises and applications

Abstract:

Yves Meyer's surprising construction of orthonormal bases consisting of dilates and translates of a single smooth function was followed soon after by the development of the Multiresolution Analysis framework in collaboration with Stephane Mallat. As already shown in the presentation by Stephane Mallat, this development was rooted in and used insights from a variety of fields -- ranging from pure harmonic analysis to statistics, quantum physics, geophysics and computer vision. The lecture will discuss some of those diverse roots in more detail, and also show how the new wavelet synthesis, sparked by Yves Meyer's seminal work, led to further progress in all those fields as well as others. Finally, hindsight shows that the new paradigm introduced by wavelet analysis was a first example of the power of sparse decompositions -- and thus a prelude to another paradigm shift, that of Compressed Sensing, about which more will follow, in the presentation by Emmanuel Candès.

**Professor Emmanuel Jean Candès, Stanford University:**

Wavelets, sparsity and its consequences

Wavelets, sparsity and its consequences

Abstract:

Soon after they were introduced, it was realized that wavelets offered representations of signals and images of interest that are far more sparse than those offered by more classical representations; for instance, Fourier series. Owing to their increased spatial localization at finer scales, wavelets prove to be better adapted to represent signals with discontinuities or transient phenomena because only a few wavelets actually interact with those discontinuities. It turns out that sparsity has extremely important consequences and this lecture will briefly discuss three vignettes. First, enhanced sparsity yields the same quality of approximation with fewer terms, a feat which has implications for lossy image compression since it roughly says that fewer bits are needed to achieve the same distortion. Second, enhanced sparsity yields superior statistical accuracy since there are fewer degrees of freedom or parameters to estimate. This gives scientists better methods to tease apart the signal from the noise. Third, enhanced sparsity has important consequences for data acquisition itself: a new technique known as compressed sensing is turning a few fields a bit upside down for it effectively says that to make a high-resolution image we need to collect far fewer samples than were thought necessary.

Closing remarks by Chair of the Abel Board, **Kristian Ranestad**

### Practical information

### 24 May 2017

10:00 - 15:30

Georg Sverdrups Hus, University of Oslo

### Abel Prize celebrations in Oslo

#### Karen Uhlenbeck received the Abel Prize from H.M. King Harald V

His Majesty King Harald V presented the 2019 Abel Prize to Karen Uhlenbeck at an award ceremony in the University Aula in Oslo on the 21st of May. Uhlenbeck is "Professor Emerita of Mathematics and Sid W. Richardson Regents Chair at the University of Texas at Austin" and "Visitor in the School of Mathematics at the Institute for Advanced Study".

(17.05.2019) More### Abel lectures at the University of Oslo

#### The lectures will be streamed

Karen Uhlenbeck gave her Abel Prize lecture on the 22nd of May at the University of Oslo. Chuu-LianTerng and Robert Bryant gave lectures related to Uhlenbeck's work. The popular science lecture was given by stand-up mathematician Matt Parker.

(14.05.2019) More### Karen Uhlenbeck first woman to win the Abel Prize

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2019 to Karen Keskulla Uhlenbeck of the University of Texas at Austin, USA “for her pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.”

His Majesty King Harald V will present the Abel Prize to the laureate at the award ceremony in Oslo on the 21^{st} of May.

### Congratulations to Karen Uhlenbeck from University of Texas at Austin

"At the University of Texas at Austin and the Department of Mathematics, we are delighted and tremendously proud of Karen Uhlenbeck, recipient of the 2019 Abel Prize" - Thomas Chen, Chair of the University of Texas at Austin Math Department

(19.03.2019) More### Congratulations to Karen Uhlenbeck from AMS President

"On behalf of the American Mathematical Society, it is my great pleasure to congratulate Professor Karen Uhlenbeck, recipient of the 2019 Abel Prize. Professor Uhlenbeck has made legendary advances in several fields of mathematics. Her early groundbreaking work on harmonic maps gave rise to a new field, geometric analysis. Her analysis via gauge theory of solutions of Yang-Mills equations, had and will continue to have a profound influence on all future work in this field. She transformed the fields of geometry and analysis, crossing boundaries and making deep discoveries at the interfaces." AMS President Jill Pipher

(19.03.2019) More