John Willard Milnor
|Alma mater||Princeton University (AB, PhD)|
|Known for||Exotic spheres|
Milnor's theorem 
Milnor-Thurston kneading theory
|Awards||Putnam Fellow (1949, 1950)|
Sloan Fellowship (1955)
Fields Medal (1962)
National Medal of Science (1967)
Leroy P. Steele Prize (1982, 2004, 2011)
Wolf Prize (1989)
Abel Prize (2011)
|Institutions||Stony Brook University|
|Doctoral advisor||Ralph Fox|
|Doctoral students||Tadatoshi Akiba |
Laurent C. Siebenmann
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the six mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize.
Milnor was born on February 20, 1931 in Orange, New Jersey. His father was J. Willard Milnor and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fary-Milnor theorem. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Robert H. Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox. His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure. Upon completing his doctorate, he went on to work at Princeton. He was a professor at the Institute for Advanced Study from 1970 to 1990.
One of his published works 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 one considers orientation).
An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. He gave a complete inventory of differentiable structures in spheres of all dimensions with Kervaire, and only continued till 2009.
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 fiber has the homotopy type of a bouquet of ? spheres where ? is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area that continues to mature to this day.
Milnor introduced the growth invariant in a finitely presented group and the theorem stating that the fundamental group of a negatively curved Riemannian manifold has exponential growth became a striking point in the foundation for modern geometric group theory, and the basis for the theory of a hyperbolic group in 1987 by Mikhail Gromov.
Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics:
It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.
His other significant contributions include microbundles, influencing the usage of Hopf algebras, algebraic K-theory, etc. He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books. He served as Vice President of the AMS in 1976-77 period.
Milnor was elected as a member of the American Academy of Arts and Sciences in 1961. 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 Lester R. Ford Award in 1970 and again in 1984, the Leroy P. Steele Prize for "Seminal Contribution to Research" (1982), the Wolf Prize in Mathematics (1989), the Leroy P. Steele Prize for Mathematical Exposition (2004), and the Leroy P. Steele Prize for Lifetime Achievement (2011) "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399-405". In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.
Milnor was awarded the 2011 Abel Prize, for his "pioneering discoveries in topology, geometry and algebra." Reacting to the award, Milnor told the New Scientist "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning." In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".
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