John Willard Milnor, (born February 20, 1931, Orange, New Jersey, U.S.), American mathematician who was awarded the Fields Medal in 1962 for his work in differential topology and the Abel Prize in 2011 for his work in topology, geometry, and algebra.
Milnor attended Princeton University (A.B., 1951; Ph.D., 1954), in New Jersey. He held an appointment at Princeton from 1954 to 1967 and, after several years at other institutions, joined the faculty at the Institute for Advanced Study, Princeton, in 1970. In 1989 he became director of the Institute for Mathematical Sciences at the State University of New York, Stony Brook.
Milnor was awarded the Fields Medal at the International Congress of Mathematicians in Stockholm in 1962. His work was part of a revival of interest in a geometric approach to topology in the 1950s. Early in the 20th century the field had been highly geometric, but in the 1930s and ’40s algebraic approaches dominated research. In particular, Milnor’s discovery of the 28 differentiable structures for the seven-dimensional sphere, S7, in 1956 was instrumental in the development of the new field of differential topology. Milnor dubbed these differentiable structures “exotic spheres.” In 1963, in collaboration with French mathematician Michel Kervaire, he calculated the number of exotic spheres for dimensions greater than 4.
Additionally, Milnor contributed to algebraic geometry on singular points of complex hypersurfaces, and in 1961 he showed that the Hauptvermutung (German: “main conjecture”), a principal conjecture in the theory of manifolds concerning triangulations of n-dimensional manifolds, which had been an open question since 1908, is not true for complexes in dimensions greater than 3. Beginning in the 1970s, he worked on complex dynamics.
Milnor was noted as an influential teacher, particularly through his books on the Morse theory and the h-cobordism theorem, which are universally regarded as models of mathematical exposition. His publications include Differential Topology (1958), Morse Theory (1963), Topology from the Differentiable Viewpoint (1965), and Dynamics in One Complex Variable (1999). His Collected Papers were published in five volumes from 1994 to 2010. He won the National Medal of Science in 1966.