Here’s a little news story I wrote for Nature on the Abel Prize. This award presents a notoriously challenging subject for science reporters each year, because it is always the devil of a job concisely to explain what on earth the recipient has done to deserve the award. I can’t deny that the same challenge applied here, but in spades, because Milnor has done so much. But it was a challenge I enjoyed. Given the choice, I’d have personally kept in the edited version the fact that holomorphic dynamics involves numbers in the complex plane, because it is the kind of thing experts will sniffily point out. But I can understand the fear that the reader will be exhausted by then. Ah, mathematics – what a wonderful, strange game it is.
John Milnor wins the ‘Nobel of maths’ for his manifold works.
Awarding Albert Einstein a Nobel prize for his research on the photoelectric effect looks in retrospect like a somewhat arbitrary choice from among the galaxy of his contributions to all of physics.
In granting the 2011 Abel Prize in mathematics to John Milnor of Stony Brook University in New York, the committee of the Norwegian Academy of Science and Letters has wisely abandoned any such attempt to single out a particular achievement. The citation states merely that Milnor has made ‘pioneering discoveries in topology, geometry and algebra’: in effect a recognition that he has contributed to modern maths across the board.
In fact, Milnor’s work goes further: it also touches on dynamical systems, game theory, group theory and number theory. In awarding this equivalent of a Nobel prize, worth around $1m, the committee states that “All of Milnor’s works display marks of great research: profound insights, vivid imagination, elements of surprise, and supreme beauty.”
His breadth is unusual, says Professor Ragni Piene of the University of Oslo, the chair of the Abel Prize committee. “Though some of the fields he has worked in are related, he really has had to learn and develop new tools and new theory.”
Milnor “says is mainly a problem solver”, adds Piene. “But in the solving process, in order to understand the problem deeply he ends up creating new theories and opening up new fields.”
Among the most surprising of Milnor’s discoveries was the existence of so-called exotic spheres, multidimensional objects with strange topological properties. In 1956 Milnor was studying the topological transformations of smooth-contoured high-dimensional shapes – that is, shapes with no sharp edges. A so-called continuous topological transformation converts one object smoothly – as though remoulding soft clay – into another, without any tears in the fabric.
He discovered that in seven-dimensions there exist smooth objects that can be converted into the 7D equivalent of spheres only via intermediates that do have sharp kinks. In other words, the only way to get from one of these smooth objects to another is by making them not smooth. Kinks and corners in a surface are said to make it non-differentiable, which means that its curvature at the kinks has no well-defined value.
These counter-intuitive exotic spheres can exist in other dimensions too. With the French mathematician Michel Kervaire, Milnor calculated that there are precisely 28 exotic spheres in seven dimensions. But there seems at first glance little rhyme or reason to the trend for other dimensions: there is just one exotic sphere in 1, 2, 3, 5 and 6 dimensions, but 992 in 11 dimensions, 1 in 12 dimensions, 16,256 in 15D, and 2 in 16D. No one has yet figured out how many there are in four dimensions. This work spawned an entire new field of mathematics, called differential topology.
Some of Milnor’s other achievements are recognizably related to such topological conundrums, such as his work on the relationships between different triangulations (representations as networks of triangles) of mathematical surfaces called manifolds. Topology was also central to some of Milnor’s earliest work in 1950 on the curvature of knots.
But his work on group theory is quite different. Group theory was partly invented by the nineteenth-century Norwegian mathematician Niels Henrik Abel, after whom the award is named. In the formulation developed by Abel, a group can be represented as all non-equivalent combinations (‘words’) of a set of symbols. Milnor and the Czech mathematician Frantisek Wolf clarified how the number of words grows as the number of symbols increases for a wide class of groups called solvable groups.
More recently, Milnor, now 80, has been working in the field of holomorphic dynamics, which concerns the trajectories generated in the plane of real and imaginary numbers by iterating equations: the branch of maths that led to the discovery of fractal patterns such as the Mandelbrot and Julia sets.
Milnor has already won just about every other key prize in mathematics, including the Fields medal (1962) and the Wolf prize (1989). But beyond his skills as a researcher, Milnor has been widely praised as a communicator. His books “have become legendary for their high quality”, according to mathematician Timothy Gowers of the University of Cambridge.