John Willard Milnor (born February 20, 1931, in Orange, New Jersey) is an American mathematician known for his work in differential topology, K-theory and dynamical systems, and for his influential books. He won the Fields Medal in 1962 and Wolf Prize in 1989. As of 2005, Milnor is a distinguished professor at the State University of New York at Stony Brook. His wife, Dusa McDuff, is a professor of mathematics at Barnard College.

Life

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 was an editor of the Annals of Mathematics for a number of years after 1962.

His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Jonathan Sondow and Michael Spivak.

Work

His most celebrated single result is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures (28 if you consider orientation). An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area which continues to develop to this day.

In 1961 Milnor disproved the Hauptvermutung by exhibiting two simplicial complexes which are homeomorphic but combinatorially distinct.

See also

* Fary–Milnor theorem
* Milnor conjecture in algebraic K-theory
* Milnor conjecture in knot theory
* Milnor conjecture concerning manifolds with nonnegative Ricci curvature
* Milnor fibration
* Milnor map
* Milnor–Thurston kneading theory
* Orbit portrait

References

Articles

* Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I". Annals of Mathematics (The Annals of Mathematics, Vol. 77, No. 3) 77 (3): 504–537. doi:10.2307/1970128. MR0148075. http://links.jstor.org/sici?sici=0003-486X%28196305%292%3A77%3A3%3C504%3AGOHSI%3E2.0.CO%3B2-R
* Milnor, John W. (1956). "On manifolds homeomorphic to the 7-sphere". Annals of Mathematics (The Annals of Mathematics, Vol. 64, No. 2) 64 (2): 399–405. doi:10.2307/1969983. MR0082103. http://links.jstor.org/sici?sici=0003-486X%28195609%292%3A64%3A2%3C399%3AOMHTT7%3E2.0.CO%3B2-3
* Milnor, John W. (1959). "Sommes de variétés différentiables et structures différentiables des sphères". Bull. Soc. Math. France 87: 439–444. MR0117744. http://www.numdam.org/item?id=BSMF_1959__87__439_0
* Milnor, John W. (1959b). "Differentiable structures on spheres". American Journal of Mathematics (American Journal of Mathematics, Vol. 81, No. 4) 81 (4): 962–972. doi:10.2307/2372998. MR0110107. http://links.jstor.org/sici?sici=0002-9327%28195910%2981%3A4%3C962%3ADSOS%3E2.0.CO%3B2-Q
* Milnor, John W. (1961). "Two complexes which are homeomorphic but combinatorially distinct". Annals of Mathematics (The Annals of Mathematics, Vol. 74, No. 3) 74 (2): 575–590. doi:10.2307/1970299. MR133127. http://www.jstor.org/pss/1970299

Books

* Milnor, John W. (1963). Morse theory. Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, NJ. ISBN 0-691-08008-9.
* Milnor, John W. (1965). Topology from the differentiable viewpoint. 1997 reprint of the 1965 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ. ISBN 0-691-04833-9.
* Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo. ISBN 0-691-08122-0.
* Milnor, John W. (1965). Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, NJ. ISBN 0-691-07996-X.
* Milnor, John W. (1968). Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo. ISBN 0-691-08065-8.
* Milnor, John W. (1999). Dynamics in one complex variable. Vieweg, Wiesbaden, Germany. ISBN 3-528-13130-6.
* Milnor, John W. (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, NJ. ISBN 978-0-691-08101-4.
* Husemoller, Dale; Milnor, John W. (1973). Symmetric bilinear forms. Springer-Verlag, New York, NY. ISBN 978-0-387-06009-5.

External links

* O'Connor, John J.; Robertson, Edmund F., "", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Milnor.html .Milnor, John
* John Milnor at the Mathematics Genealogy Project
* Home page at SUNYSB
* Photo