John Milnor was educated at Princeton University, receiving his A.B. in 1951. He began research at Princeton after graduating and, in 1953 before completing his doctoral studies, he was appointed to the faculty in Princeton.
In 1954 Milnor received his doctorate for his thesis Isotopy of Links written under Ralph Fox's supervision. Milnor remained on the staff at Princeton where he was an Alfred P Sloan fellow from 1955 until 1959. He was promoted to professor in 1960 then, in 1962, Milnor was appointed to the Henry Putman chair.
Milnor was awarded a Fields Medal at the 1962 International Congress of Mathematicians in Stockholm. His most remarkable achievement, which played a major role in the award of the Fields Medal, was his proof that a 7-dimensional sphere can have several differential structures. This work opened up the new field of differential topology.
Milnor showed that 28 different differentiable structures exist on the seven-dimensional sphere. He distinguished between these structures using numerical invariants based on the Todd polynomials. The Todd polynomials were first studied in algebraic geometry and it is surprising that they play this fundamental role in classification of manifolds. The reason that Milnor could use them to distinguish the differential properties of manifolds is because they have arithmetic properties, involving the Bernoulli numbers, which reflect in a deep and not fully understood way these differential properties.
The references  to  give a good indication of the wide influence of Milnor's work up to 1992 (when these articles were written). The article  is a survey of Milnor's work in algebra, particularly in algebraic K-theory, where his work continues to have important influences.
The article  looks at nine papers which Milnor had written on differential geometry. It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4π. Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group.
In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in . He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. He also studied the Steenrod algebra and its dual, investigated the structure of Hopf algebras, and studied characteristic classes and their relation to mathematical physics.
Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarised by Peter Makienko in his review of :-
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.
Milnor has received many awards for work. He received the National Medal of Science in 1967 and was elected a member of the National Academy of Sciences, the American Academy of Arts and Science. He is a member of the American Philosophy Society and has played a major role in the American Mathematical Society. In August 1982 Milnor received the Leroy P Steele Prize:-
...for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399-405.
Among the many services he has rendered to mathematics is editorial work, being editor of the Annals of Mathematics from 1962. Since 1988 he has been at the State University of New York at Stony Brook.
Article by: J J O'Connor and E F Robertson
Click on this link to see a list of the Glossary entries for this page