David Mumford was born on June 11, 1937 in Three Bridges (Sussex, United Kingdom) to an English father and American mother. He thinks his talent for mathematics came from his paternal grandmother, who studied math at the University of Cambridge.
In 1940 his family moved to the United States, and in 1953 he entered Harvard University to study pure mathematics. Among his teachers there was future mentor Oscar Zariski, who would encourage his interest in algebraic geometry. After completing a doctorate at Harvard in 1961, he stayed on in a series of posts before taking up the chair of the Department of Mathematics in 1981. It was during this period that he received the Fields Medal (1974) for his achievements in the field of algebraic geometry.
By the early 1980s, Mumford had turned his attention to computing and brain processes, with the ultimate ambition of developing a mathematic model capable of simulating human thought. He joined the Division of Applied Science at his alma mater in 1985 then, eleven years later, moved to the Division of Applied Math at Brown University, attracted by its interdisciplinary culture. It was here that he conducted his pathbreaking work in the application of mathematics to computer vision. He has supervised around fifty PhD theses.
He was awarded the Wolf Prize in Mathematics in 2008 and two years later received the U.S. National Medal of Science, the highest scientific honor the country can bestow.
Speech
Basic Sciences, 5th edition
What does the working of the human brain have in common with works of art, computer vision, algebraic geometry and theoretical physics? Mathematics, of course. For David Mumford and Ingrid Daubechies, joint winners of the BBVA Foundation Frontiers of Knowledge Award in Basic Sciences, mathematics is a universal tool with which to describe – and discover – the world. A DNA of all the sciences whose very ubiquity is what has enabled them to build such extraordinarily varied careers. Mumford, at a young age, “was entranced” by the abstraction of pure math, but then decided to study how our thoughts are generated and went on from there to the field of computer vision. Daubechies started out as a theoretical physicist, and ended up developing a powerful tool for the analysis of images. Her work is used in mobile phones, medical diagnostics and the evaluation of works of art.
In distinguishing Daubechies (Houthalen, Belgium; 1954), the jury singled out her work on a mathematical tool called wavelets, “which resulted in a new approach to data compression, with a strong impact on a multitude of technologies.” Mumford was cited for “his contributions to algebraic geometry and to the mathematics of computer vision.”
If mathematics underlies just about everything, perhaps its psychological equivalent is curiosity. Mumford recalls that he always wanted to “explore the world,” and even today his interests range from neurobiology to the history of math by way of education and Middle East politics. Daubechies too talks about how, as a child, she had to know how everything worked: she was astounded at the rapid advance of the powers of two and avid to know why a car’s headlights at night would seem to drain color from the objects they illuminated. Nowadays, both scientists may be set off in new research directions by a conversation with experts in another field. Hence she alludes to him as one of her “mathematical heroes,” while he talks admiringly about her incursions into the world of art: “It’s wonderful to see a rather abstract mathematical theory being used in such an unexpected way.”
What is the secret of the mathematical gaze? Strangely – though perhaps not entirely – both settle on the same characteristic: mathematics looks for patterns. “A mathematician has a unique perspective in that he or she seeks a simple mathematical model that captures some of the essential features of a real-world situation,” says Mumford. And for Daubechies: “Mathematics is the right tool when you intuit that there is a structure there and you want to put your finger on it.” That structure may live in the world of the abstract or in concrete reality. Mumford and Daubechies have pursued it in both.
Mumford, an emeritus professor at Brown University (United States), discovered that “the abstraction [of pure math] was very seductive” when studying under Oscar Zariski at Harvard University, of whom he says: “He made algebraic geometry seem like magic.” Mumford’s work in this area over the next 25 years was successful enough to earn him the Fields Medal in 1974. But he had always felt drawn to applications – even working as an undergraduate on the atomic reactor of a submarine – and in the 1980s decided to turn back to the problem that had interested him since adolescence: finding a model for the working of the human brain.
He contends that our brain operates statistically, storing massive amounts of information and checking it constantly against what it perceives right now. For example: “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 the engine of a truck.” Mumford has applied this model to computer vision. His work in the field has not led to specific industrial developments, but is at the heart of applications such as driverless cars.
He acknowledges though that knowing the brain in its entirety is too ambitious a goal: “One needs experiments that reveal how groups of neurons work together, what sort of ‘cell assemblies’ they make on the fly to instantiate a thought. There are so many possibilities here that we would need to record millions of neurons, and that is not yet possible.”
Daubechies, a professor at Duke University (United States) earned her PhD in theoretical physics. But for her the move into mathematics has always been a natural one: “I realized that the math I was developing for physics could be useful to other scientists, and that is how I came to switch areas,” she recalls. Also, although her work on wavelets has taken her more to the applied side, she refuses to forego “the beauty that resides in the purest mathematics.” Wavelets are a mathematical tool that, among other things, allows us to compress and retrieve data with almost no loss of information. They have been used to develop the JPEG 2000 image compression standard and to codify the FBI fingerprint database. When we upload a mobile photo to some social network, it is Daubechies’ invention that does the job. But her own interest is increasingly focused on the analysis of paintings by Van Gogh, Gauguin and the Flemish artists: for wavelets, it turns out, can not only confirm their authenticity, but can also reveal how the great masters worked by analyzing the sketches beneath the paint.
Mumford and Daubechies’ careers also intersect at other points. He was president of the International Mathematical Union (IMU) from 1994 to 1998, and she has held the same position since 2010. Both are concerned about the teaching of mathematics. Daubechies is keen to puncture the air of esotericism that hangs around the science: “We all use mathematical reasoning in our daily lives”; Mumford, its reputation as a cold discipline: “Mathematics is just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty.”
