# John W. Milnor

Wolf Prize Laureate in Mathematics 1989

**John W. Milnor**

#### Affiliation at the time of the award:

**Institute for Advanced Study****, USA**

#### Award citation:

**“for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint”.**

#### Prize share:

**John W. Milnor **

**Alberto P. Calderon**

During his undergraduate years at Princeton University, John W. Milnor (born in 1931, USA) was named a Putnam Fellow in both 1949 and 1950. Notably, at the youthful age of 19, he successfully proved the Fáry–Milnor theorem. Upon completing his studies, Milnor earned an A.B. in mathematics in 1951, presenting a senior thesis entitled “Link groups,” which was supervised by Robert H. Fox. Remaining at Princeton for his graduate studies, Milnor continued his academic pursuit and obtained his Ph.D. in mathematics in 1954. His doctoral dissertation, titled “Isotopy of links,” was also conducted under the guidance of Fox.

The highly original discoveries of Professor John W. Milnor in geometry have exerted a major influence on the development of contemporary mathematics. The current state of the classification of topological, piecewise linear, and differentiable manifolds rests in large measure on his work in topology and algebra.

Milnor’s discovery of differentiable structures on S7 (the 7-dimensional sphere) which are exotic, i.e. different from the standard structure, came as a complete surprise and marked the beginning of differential topology. Later, in joint work with Kervaire, Milnor turned these structures (on any Sn) into a group which could then (in part) be computed; it turns out that there are over sixteen million distinct differentiable structures on S31 In his important work in algebraic geometry on singular points of complex hypersurfaces, exotic spheres are related to links around singularities. In the combinatorial direction, Milnor disproved the long-standing conjecture of algebraic topology known as the Hauptvermutung, by constructing spaces with two polyhedral structures that cannot have a common subdivision. This was based on an unexpected use of the previously known concept of torsion, which has since become, in its various algebraic and geometric versions, a basic tool.

These are just some highlights of Milnor’s impressive body of work. Beyond the research papers, a wealth of new results are contained in his books. These are famous for their clarity and elegance and remain a source of continuing inspiration.