Chercheurs

Mathématiques / MILNOR, JOHN W.

jack[a]

Position

Professor of Mathematics
Co-Director, Institute for Mathematical Sciences
Stony Brook University, Stony Brook NY, USA.

Thèmes de recherche

His most celebrated single result is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures (28 if you consider orientation). An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fibre has the homotopy type of a bouquet of ? spheres where ? is known as the Milnor number. Milnor\'s 1968 book on his theory inspired the growth of a huge and rich research area which continues to develop to this day.

In 1961 Milnor disproved the Hauptvermutung by exhibiting two simplicial complexes which are homeomorphic but combinatorially distinct.

Prix et récompenses

Fields Medal (1962)
BMC morning speaker (1964)
Hedrick lecturer (1965)
AMS Colloquium Lecturer (1968)
AMS Steele Prize (1982)
Wolf Prize (1989)
AMS Steele Prize (2004)

Publications

Books

Morse Theory (with M. Spivak and R. Wells), Ann. Math. Studies 51, Princeton U. Press
1963. (Russian translation 1965; Japanese translation 1968; Korean translation 1999.)

Lectures on the h-Cobordism Theorem (with L. Siebenmann and J. Sondow), Princeton
U. Press 1965. (Russian translation 1969; revised edition in preparation 2007.)

Topology from the Differentiable Viewpoint, U. Virginia Press 1965, Princeton University
Press 1998. (Russian translation 1972; as part of “Differential Topology: a First Course”;
Japanese translation 1998.)

Singular Points of Complex Hypersurfaces, Ann. Math. Studies 61, Princeton U. Press
1968. (Russian translation 1971; Japanese translation 2003.)

Introduction to Algebraic K-Theory, Ann. Math. Studies 72, Princeton U. Press 1971.
(Russian translation 1974.)

Symmetric Bilnear Forms (with D. Husemoller), Springer 1973. (Russian translation
1986.)

Characteristic Classes (with J. Stasheff), Princeton U. Press 1974. (Russian translation
1979; Japanese translation 2001; reprinted Hindustan Book Agency 2005.)

Collected Papers I, Geometry, Publish or Perish, AMS 1994.

Collected Papers II, The Fundamental Group, Publish or Perish, AMS 1995.

Dynamics in One Complex Variable, Vieweg 1999, 2000; Princeton U. Press 2006.

Collected Papers III, Differential Topology, AMS 2007.

Collected Papers IV, Homotopy, Homology and Manifolds, (J. McCleary editor), AMS
2009.

Collected Papers V, Algebra, (H. Bass and T.-Y. Lam editors), AMS 2010.

Papers

Local connectivity of Julia sets: expository lectures, pp. 67-116 of “The Mandelbrot
set, Theme and Variations” edit: Tan Lei, LMS Lecture Note Series 274 , Cambr. U.
Press 2000.

Rational maps with two critical points, Experimental Math. 9 (2000) 481-522.

On Entropy and Monotonicity for Real Cubic Maps (with Tresser; and with an appendix
by Douady and Sentinac) Comm. Math. Phys. 209 (2000) 123-178.

Classification of (n-1)-connected 2n-manifolds, and the discovery of exotic spheres, pp.
25-30 of “Surveys on Surgery Theory 1”, edit. Cappell, Ranicki and Rosenberg, Ann.
Math. Stud. 145, Princeton U. Press 2000.

Periodic orbits, external rays and the Mandelbrot set: An expository account, Asterisque
261 (2000) “Geometrie Complexe et Systemes Dynamiques”, pp. 277-333.

Towards the Poincare conjecture and the classification of 3-manifolds, Notices A.M.S.,
November 2003

Pasting together Julia sets; a worked out example of mating, Experimental Math 13
(2004) 55-92.

On Latt`es maps, in “Dynamics on the Riemann Sphere, A Bodil Branner Festschrift”,
edit. Hjorth and Petersen, Eur. Math. Soc 2006, pp. 9-43.

Elliptic curves as attractors in P2 : Part 1 Dynamics (with A. Bonifant and M. Dabija);
Experimental Math. 16 (2007) 385-420.

Schwarzian derivatives and cylinder maps (with A. Bonifant); in Holomorphic dynamics
and renormalization, Fields Inst. Commun., 53, Amer. Math. Soc., Providence, RI, 2008,
pp. 1-21.

Fifty years ago: Topology of manifolds in the 50’s and 60’s; IAS/Park City Mathematics
Series 15, “Low Dimensional Topology”, edit. T. Mrowka and P. Ozsv´ath, AMS 2009, pp.
9-20.

Cubic polynomial maps with periodic critical orbit I, in “Complex Dynamics, Families
and Friends”, edit. D. Schleicher, A.K. Peters 2009, pp. 333-411.

Cubic polynomial maps with periodic critical orbit II: Escape regions (with A. Bonifant
and J. Kiwi), ArXiv:0910.1866, 2009, Conform. Geom. Dyn. 14 (2010), 68-112 and 190-
193.

Cubic polynomial maps with periodic critical orbit III: External rays (with A. Bonifant),
in preparation.