John Milnor

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John Willard Milnor (b. February 20, 1931) is a mathematician known for his work in differential topology and K-theory and his influential books, which are widely considered to be examples of fine mathematical writing. He won the Fields Medal in 1962. As of 2005, John Milnor is a distinguished professor at the State University of New York at Stony Brook with his wife Dusa McDuff.

As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fary-Milnor theorem. He continued on to graduate school at Princeton and wrote his thesis, entitled isotopy of links, which concerned link groups (a generalization of the classical knot group) and their associated link structure. His advisor was Ralph Fox. Upon completing his doctorate he went on to work at Princeton.

In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He later went on to win the National Medal of Science (1967), the Leroy P Steele Prize for Seminal Contribution to Research (1982), the Wolf Prize in Mathematics (1989), and the Leroy P Steele Prize for Mathematical Exposition (2004).

He has been editor of the Annals of Mathematics since 1962. His books include Topology from the Differentiable Viewpoint, Morse Theory, Characteristic Classes (with Stasheff), The h-Cobordism Theorem, Dynamics in One Complex Variable, and Singular Points of Complex Hypersurfaces.

His most celebrated single result is his proof of existence of 7-spheres with nonstandard differential structure. Later with Kervaire, he showed the 7-sphere has 15 differentiable structures (28 if you consider orientation). An n-sphere with nonstandard differential structure is called an exotic sphere.

His students have included Michael Spivak, John Mather, and Tadatoshi Akiba.

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