John Willard Milnor

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John Willard Milnor, 2007

John Willard Milnor (born February 20, 1931 in Orange , New Jersey ) is an American mathematician . He currently teaches mathematics as a professor at the State University of New York at Stony Brook in New York and is co-director at the Institute for Mathematical Sciences there .


Milnor is the son of an engineer. He studied at Princeton University , where he also received his PhD from Ralph Fox in 1954 (on "link groups", which generalize node groups). In 1949, while still a student, he proved the Fáry and Milnor theorem , which states that a space curve is an unknot if the integral of the curvature along the closed curve is ≤ 4π. He solved a conjecture made by Karol Borsuk in 1947 while he was a student of Albert W. Tucker . Borsuk and independently Werner Fenchel had proven that the total curvature of a closed space curve is always greater than or equal to 2π, whereby the equality only applies if the curve borders a flat convex area. Borsuk then asked if there were lower limits for the curvature of knotted curves. Since his student days, Milnor was friends with John Nash , with whom he began to deal with game theory and whom he helped in later years to find a job after his illness.

In 1960 he became a professor of mathematics at Princeton and took over the chair in 1962. In the same year he was awarded the Fields Medal at the International Congress of Mathematicians in Stockholm for his proof that different differentiable structures can exist on the 7-dimensional sphere , it is a so-called " exotic sphere ". With Michel Kervaire he showed that there are exactly 15, taking into account the orientation 28. Milnor also dealt with the topology of singularities , in which the exotic spheres also play a role (inter alia Milnor fibers ).

In 1961 he found first indications for counterexamples (in dimension 6) to the so-called main conjecture (by Heinrich Tietze ) about the uniqueness of the triangulability of topological manifolds. In 1964 he showed that the eigenvalue spectrum of the Laplace operator is not sufficient to characterize compact Riemannian manifolds down to isometry (his counter-example was two 16-dimensional tori ). For surfaces , this led to the Can one hear the shape of a drum? Problem by Mark Kac .

He also wrote reports on game theory for the Rand Corporation , including a. 1951 Games against nature , which also deals with quantum mechanics . In 1954, Sum of positional games was the first study of non-neutral games in combinatorial game theory .

Milnor's books on algebraic topology and differential topology (often only hectographed ) are considered standard works.

In addition to his work on differential topology, he contributed significantly to the development of the algebraic K-theory . Another area of ​​interest of Milnor is dynamics , especially holomorphic dynamics ( iteration of holomorphic functions).

He is married to the topologist Dusa McDuff .

His students include John N. Mather , Jonathan Sondow, Michael Spivak and Laurent Siebenmann .


Milnor received the following prizes and honors for his work:


  • Lisa Goldberg, Anthony Phillips (eds.): Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor's 60th Birthday . Publish or Perish 1993

from Milnor:

  • Morse theory . Princeton 1963 (Derivation of Bott periodicity theorem in stable homotopy)
  • Topology from the differentiable viewpoint . Princeton 1965, 1997 (first Charlottesville, University of Virginia)
  • with James D. Stasheff : Characteristic classes . Princeton 1974
  • Lectures on the h-cobordism theorem . Princeton 1965
  • Singular points of complex hypersurfaces . Princeton 1968
  • Introduction to algebraic K-theory . Princeton 1971
  • Differential topology . AMS, 2007
  • with Dale Husemöller : Symmetric bilinear forms . Springer, 1973
  • History of hyperbolic geometry . In: Bulletin AMS , 1982
  • On manifolds homeomorphic to the seven sphere . In: Annals of Mathematics , 2nd series, Volume 64, 1956, p. 399
  • with Michel Kervaire Groups of homotopy spheres . In: Annals of Mathematics , Volume 77, 1963, p. 504
  • with Raoul Bott On the parallelizability of the spheres . In: Bulletin AMS , 1958
  • Survey of cobordism theory . In: L'Enseignment Mathematique , 1962
  • Spin structures on manifolds . In: L'Enseignment Mathematique , 1962
  • Eigenvalues ​​of the Laplacian on certain manifolds . In: Proc. Nat. Acad. USA , Volume 51, 1964, p. 542, (PDF)
  • Analytic proofs of the hairy ball theorem and the Brouwer fixed point theorem . In: American Mathematical Monthly , August 1978, p. 521
  • Dynamics of one complex variable . 3. Edition. Princeton 2006; vieweg, 2000
  • Games against nature . Rand Corporation, 1951 ( , PDF; 748 kB), also in Thrall et al. a .: Decision processes . New York 1954
  • Sum of positional games . In: Contrib. Theory of Games , II. In: Ann. Math. Stud. , 28, 1953, pp. 291–301 ( abstract in Zentralblatt MATH)
  • Milnor: Periodic orbits, external rays and the Mandelbrot set - an expository account . 1999, arxiv : math.DS / 9905169
  • Milnor: Dynamics of 1 complex variable- Lectures . 1992, arxiv : math.DS / 9201272
  • Milnor: Differential topology 46 years later . In: Notices AMS , 2011, No. 6

See also

Web links

Individual evidence

  1. John Willard Milnor in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. Milnor: On the total curvature of knots . In: Annals of Mathematics , Volume 52, 1950, pp. 248-257. Istvan Fary independently proved the sentence in France, Bull. ( Memento of the original from January 19, 2012 in the Internet Archive ) Info: The archive link was inserted automatically and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. SMF, Volume 77, 1949, p. 129. The anecdote that Milnor used it to solve a problem accidentally posed as homework seems to be a legend, see Mathoverflow, quoted from an email from Milnor . There was a similar anecdote about George Dantzig . @1@ 2Template: Webachiv / IABot /
  3. Milnor, Two complexes which are homeomorphic but combinatorially distinct, Annals of Mathematics, Volume 74, 1961, pp. 575-590
  4. ^ John W. Milnor: Sums of positional games . In: Contributions to the Theory of Games , Volume II. In: Annals of Mathematics Studies , 28, 1953, pp. 291–301, doi: 10.1515 / 9781400881970-017 ( abstract in Zentralblatt MATH)
  5. ^ Member History: John W. Milnor. American Philosophical Society, accessed October 31, 2018 .