Milnor was born on February 20, 1931, in Orange, New Jersey.[1] His father was J. Willard Milnor, an engineer,[2] and his mother was Emily Cox Milnor.[3][4] As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950[5] and also proved the Fáry–Milnor theorem when he was only 19 years old. 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.[6] 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.[7] His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure, classifying Brunnian links up to link-homotopy and introduced new invariants of it, called Milnor invariants. 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.
He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books which are famous for their clarity, presentation, and an inspiration for the research by many mathematicians in their areas even after many decades since their publication. He served as Vice President of the AMS in 1976–77 period.
One of Milnor's best-known works is his proof in 1956 of the existence of 7-dimensionalspheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term exotic sphere, referring to any n-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres, showing in particular that the 7-sphere has 15 distinct differentiable structures (28 if one considers orientation).
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 μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory, Singular Points of Complex Hypersurfaces, inspired the growth of a huge and rich research area that continues to mature to this day.
In 1984 Milnor introduced a definition of attractor.[8] The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors.
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.[9]
Milnor was awarded the 2011 Abel Prize,[16] for his "pioneering discoveries in topology, geometry and algebra."[17] 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."[18]
In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".[19]
—— (1965). Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN0-691-07996-X. OCLC58324.
—— (1968). Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN0-691-08065-8.
—— (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN978-0-691-08101-4.
Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN0-691-08122-0.[22]
Milnor, John W. (1997) [1965]. Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN0-691-04833-9.
—— (1999). Dynamics in one complex variable. Wiesbaden, Germany: Vieweg. ISBN3-528-13130-6.2nd edn. 2000.[23]
—— (1961). "Two complexes which are homeomorphic but combinatorially distinct". Annals of Mathematics. 74 (2). Princeton University Press: 575–590. doi:10.2307/1970299. JSTOR1970299. MR0133127.
^Lyubich, Mikhail (1993). "Back to the origin: Milnor's program in dynamics". In Goldberg, Lisa R.; Phillips, Anthony Valiant (eds.). Topological Methods in Modern Mathematics: A Symposium in Honor of John Milnor's Sixtieth Birthday. Publish or Perish. pp. 85–92. ISBN0-914098-26-8.
^Goldberg, Lisa R.; Phillips, Anthony V., eds. (1993), Topological methods in modern mathematics, Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991, Houston, TX: Publish-or-Perish Press, ISBN978-0-914098-26-3
^"2011: John Milnor". Abelprisen (Abel Prize) website. Retrieved August 22, 2022.
"Seminar Videos, IMS Video Collection". Institute for Mathematical Sciences, Stony Brook University. (40 links from 1965 to May 2021, with 9 videos from Milnor's seminars)
