The two have formulated solutions to varied and complex problems starting from the vantage point of pure mathematics, but guided by a multidisciplinary approach. Curiously, however, their paths have run in opposite directions: while Daubechies trained as a physicist but ended up in mathematics, Mumford’s mathematical research has had considerable influence among theoretical physicists.

The jury singled out Daubechies (Houthalen, Belgium, 1954), a professor at Duke University (United States), for her work on “wavelets, which resulted in a new approach to data compression, with a strong impact on a multitude of technologies, including efficient audio and video transmission and medical imaging.”

Mumford (Sussex, United Kingdom,1937), Emeritus Professor at Brown University (United States), is recognized particularly for “his contributions to algebraic geometry and to the mathematics of computer vision. He has applied tools of variational calculus to the theory of vision and developed statistical models for imaging and pattern recognition. His work has had a lasting impact in both pure and applied mathematics.”

Mumford, who declared himself “pleased and surprised” by the jury’s decision, explained that he was drawn to pure mathematics by a teacher who “made the field of algebraic geometry seem almost magical.” His own work, which he describes as “making maps” that offer “a bird’s eye view” of algebraic geometry, was impressive enough to earn him the Fields Medal in 1974, aged just 37.

Mapping the brain mathematically

As early as adolescence, Mumford knew that he wanted to explore the workings of the human brain, and after his successes in pure mathematics he was clear that “the time had come to make a change.” So after leading the field of algebraic geometry for all of 25 years, he turned in the 1980s to a new problem: how to mathematically render the human ability to understand an image. “We take for granted the fact that when we enter a strange room, we immediately understand the layout and things we see there, but when people tried to get a computer to do that they discovered it was tremendously hard.”

One of his insights was that the brain operates by combining previous knowledge with what it is perceiving right now: “If I am walking in Boston and hear something like a growl, I know that it is unlikely to be a tiger, so I recognize, for instance, the engine of a truck.” His mission, now being carried forward by his students, is to describe this human ability in mathematical terms.

It is the sheer sweep of Mumford’s work that is singled out by his nominator Antonio Campillo, President of the Real Sociedad Matemática Española (RSME): “David Mumford represents all mathematical sciences, and all researchers in mathematics, independently of their field of interest in pure, applied, computational or other areas of mathematics.”

A physicist drawn to the beauty of mathematics

Ingrid Daubechies is open in her delight at sharing this award with Mumford. She is a theoretical physicist and, as such, her career began a long way from the signal processing math for which she is now being distinguished. Her move into mathematics was a product of theoretical physics’ urgent need for new mathematical tools and models. In her case too, the combination of basic research and applications came about naturally.

Indeed, although her work on wavelets has taken her towards applications, Daubechies refuses to forego the beauty of pure mathematics: “I wanted to preserve all that beauty and be practical at the same time.”

Wavelets are a tool that allows to deconstruct a mathematical object, or an image, for example, into simpler components. In practical terms, this deconstruction means we can transmit information-rich images with no loss of quality: “This decomposition into building blocks makes it possible for a soccer
game to be transmitted and for the players to be seen really sharply, though maybe not the grass where you don’t care so much.”

Daubechies’ work on wavelets has one of its best known applications in the JPEG 2000 image compression standard, but it is also a powerful instrument for testing theorems in the ambit of basic research in pure mathematics.

Another point of contact between Mumford and Daubechies is their love of interdisciplinarity. Not for nothing does Mumford work in a field – computer vision – where “mathematics are just one small part; you also have engineers, neuroscientists…” And Daubechies will happily turn her mind to problems in other areas such as art: “Someone drew my attention to the fact that image analysis can be used to distinguish an artist’s brushstroke,” so we can tell, for instance, whether a painting is authentic.

She is now exploring this with art historians, a development that fascinates Mumford: “It’s wonderful to see a rather abstract mathematical theory being used in such an unexpected way.”

Daubechies’ nomination comes from another Spaniard, Manuel de León, a Research Professor with CSIC and Director of its Institute of Mathematical Sciences (ICMAT). León calls her work “a clear example of the power of mathematics,” since “anyone who uses computers, internet or digital imaging benefits directly from her achievements.”

Mumford and Daubechies have both given proof of their commitment to the mathematical community, as presidents of the International Mathematical Union (IMU). Daubechies has occupied this position since 2010, while Mumford served from 1994 to 1998.

Daubechies stated her main goals on taking up the presidency as to contribute to the growth of mathematics in both developing and rich countries and to improve math teaching at all levels: “Children in high school say they hate mathematics, but what they really hate is the way it is presented to them. Because mathematics is really just common sense thinking but pushed further. Math explanations start with saying obvious things, then you progress in little steps to get to a non-obvious result.”

International jury

The jury in this category was chaired by Theodor W. Hänsch, Head of the Division of Laser Spectroscopy at the Max Planck Institute of Quantum Optics (Germany) and Nobel Physics laureate, with Avelino Corma, Research Professor in the Department of Catalysis of the Instituto de Tecnología Química, CSIC-Universidad Politécnica de Valencia (Spain) acting as secretary. Remaining members were Douglas Abraham, Professor of Statistical Mechanics in the Rudolf Peierls Centre for Theoretical Physics at the University of Oxford (United Kingdom); Ignacio Cirac, Director of the Theory Division at the Max Planck Institute of Quantum Optics (Germany); Trevor Hastie, Professor of Statistics and of Health, Research and Policy at Stanford University (United States); Nigel Hitchin, Savilian Professor of Geometry at the University of Oxford (United Kingdom); Hongkun Park, Professor of Chemistry and Chemical Biology and of Physics at Harvard University (United States); Martin Quack, Professor of Physical Chemistry at ETH Zurich (Switzerland); and Sandip Tiwari, Charles N. Mellowes Professor of Engineering at Cornell University and Director of the National Nanotechnology Infrastructure Network.