Pioneer of High-Dimensional Spaces Wins Abel Prize

Today the Norwegian Academy of Science and Letters announced that the Abel Prize in mathematics for 2011 will be awarded to John Milnor, a topologist and dynamical systems theorist at Stony Brook University in New York state. King Harald of Norway will present the prize, which includes a monetary award of approximately $1 million, in a ceremony in Oslo on 24 May. The Abel Prize, first awarded in 2003, has rapidly acquired a nearly Nobel-level cachet among mathematicians.

For Milnor, the prize caps a long and distinguished mathematical career. He first attracted attention in 1950, when, as an undergraduate at Princeton University, he solved a previously unsolved problem on the total curvature of knots. Three years later he was appointed to the math faculty at Princeton while still working on his doctorate.

In 1956, Milnor produced a result that other mathematicians immediately recognized as a masterpiece for the ages: proving the existence of “exotic” 7-dimensional spheres. By the rules of the mathematical field of topology, two spaces are considered equivalent if one can be bent, stretched, and perhaps folded until it looks like the other. Creases are not allowed. Before Milnor, no one had any inkling that this restriction made any difference; for spaces of three dimensions or fewer, it does not. (If you can pack a shirt into your suitcase with creases, you can do it without.) But Milnor constructed a seven-dimensional sphere (in fact, exactly 28 of them) that is too badly twisted to be unscrambled without creating corners and folds. This was the first sign of a dramatic difference between “smooth” spaces and “continuous” spaces in more than three dimensions. The dichotomy has led to decades of research in mathematics and physics, including the discovery of “fake” four-dimensional spaces by Simon Donaldson in 1984.

Milnor is also known as a great expositor, whose books are low on abstruseness and high on simple, common-sense explanations. “His book on Morse theory is a model of mathematical clarity, the best book I have ever read,” says Douglas Ravenel of the University of Rochester in New York state. “We are teaching a seminar on it 50 years later.” John McCleary of Vassar College in Poughkeepsie, New York, who has edited Milnor’s collected works, calls Milnor a master of understated elegance. “Reading a paper by him is like wandering into an old, old church. You wander off to one side, into the nave, and suddenly you come across this spectacular altar, or a beautiful painting that was totally unexpected.”

This is a feeling that is shared by the master himself. Milnor says that what he loves most about mathematics is “a feeling of miracles.” He adds, “You’re working on a problem and it seems impossibly hard, but then you just put together an idea here and an idea there, and somehow the answer just drops out.”

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