Month: March 2011

John Milnor and His Spherical Exotica

John Milnor’s discovery of exotic smooth spheres unleashed the exploration of new mathematical riches. He was just awarded the 2011 Abel prize. Credit: MFO via Creative Commons License

On March 23, the Norwegian Academy of Science and Letters announced that it was awarding the 2011 Abel Prize — Norway’s mathematical answer to Sweden’s Nobels — to John Milnor. The Stony Brook mathematician, 80, had already won a Fields Medal and nearly every other major math prize there is.

As most math grad students have done over the past half-century — at least those students specializing in topology and geometry — when I was in grad school I learned much of what I know from Milnor’s canonized textbooks. (The most standard ones are probably “Characteristic Classes,” which he co-authored with James Stasheff, and “Morse Theory,” but my favorite one was “Lectures on the h-Cobordism Theorem,” a typewritten gem that has long been out of print but can be found online if you google it. Some day I’ll blog about why I liked it so much.)

And perhaps as many other topology students have done over the years, I have often tried to imagine what it must have felt to be a topologist in 1956, when Milnor announced his discovery of a 7-dimensional sphere that was homeomorphic, but not diffeomorphic, to the standard 7-dimensional sphere.

What does that mean? I have tried my best to explain the basic idea in my Scientific American article on the solution of the Kervaire invariant problem and then again in the article I co-wrote with my colleague John Matson on the announcement of Milnor’s Abel prize for ScientificAmerican.com.