# John Willard Milnor

# John Willard Milnor

**1931-**

**American Mathematician**

In 1962 mathematician John Willard Milnor was awarded a Fields Medal for his work in differential topology, including his proof that a seven dimensional sphere could have 28 differential structures. In addition to the prestigious Fields Medal, Milnor has received many other honors, including the National Medal of Science in 1967.

In 1989 Milnor received the Wolf prize, an international prize intended to promote science and art for the benefit of mankind. The Wolf Foundation praised Milnor "for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint."

Milnor was born in New Jersey and undertook his undergraduate and graduate studies at Princeton, completing his doctoral thesis on the "Isotopy of Links" in 1954. Milnor displayed such exceptional brilliance as a student that he was offered an appointment to the Princeton faculty prior to the actual completion of his studies. After holding several academic positions, including a faculty appointment at UCLA, in 1970 Milnor was invited to join the Institute for Advanced Study at Princeton. In 1989 he took over as director of the Institute for Mathematical Sciences at the State University of New York in Stony Brook.

Milnor has served as vice-president of the American Mathematical Society (1975-76), editor of the *Annals of Mathematics,* and as an American Mathematical Society Colloquium Lecturer in 1968. He was also elected as a member of the National Academy of Science and the American Academy of Arts and Science.

Milnor is often credited with returning a geometrical emphasis to topological mathematics. Prior to Milnor there had been an increasing move toward using algebraic approaches to define and solve problems in topology. Milnor's 1956 publication on multiple differential structures for seven-dimensional spheres was considered a milestone in the return to geometric approaches in differential topology. Differential topology encompasses the study of differential manifolds, differentiable maps, and vector fields, among other topics, and utilizes qualitative concepts originally designed for differential geometry and differential calculus. Basic concepts of differential topology rely on the notion of a jet, embedded manifolds and intrinsic manifolds, manifold boundaries, and isomorphisms of differentiable structures. Problems and methodologies concern differential topology, differentiable manifolds, morphisms, cobordant manifolds, and what are termed surgery techniques.

Milnor's work ranges over—but is certainly not limited to—contributions in algebraic K-theory, differential geometry, algebraic topology, total curvature, duality theorems, differentiable structures, Betti numbers, polylogarithms, n-person games, and directional entropies. In 1961 Milnor provided important clarifications of the limitations of certain theories regarding manifolds. Milnor's work in the late 1990s included research in dynamics, especially holomorphic dynamics, and the relationship of low-dimensional dynamics to systems theory, properties of rational maps, periodic orbits, and Mandelbrot sets.

Regarding his approach to mathematics, Milnor is quoted as saying: "If I can give an abstract proof of something, I'm reasonably happy. But if I can get a concrete, computational proof and actually produce numbers I'm much happier. I'm rather an addict of doing things on the computer, because that gives you an explicit criterion of what's going on. I have a visual way of thinking, and I'm happy if I can see a picture of what I'm working with."

Milnor's advancements have allowed for refined conceptualization of curved space-time demanded by general relativity theory. Milnor has also advanced what is regarded as Milnor's theorem regarding the curvature of a knot.

**K. LEE LERNER**

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**John Willard Milnor**