# John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the six mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize.

John Willard Milnor
BornFebruary 20, 1931 (age 89)
NationalityAmerican
Alma materPrinceton University (AB, PhD)
Known forExotic spheres
Fary–Milnor theorem
Hauptvermutung
Milnor K-theory
Microbundle
Milnor Map
Milnor's theorem [1]
Surgery theory
Spouse(s)Dusa McDuff
AwardsPutnam Fellow (1949, 1950)
Sloan Fellowship (1955)
Fields Medal (1962)
National Medal of Science (1967)
Leroy P. Steele Prize (1982, 2004, 2011)
Wolf Prize (1989)
Abel Prize (2011)
Scientific career
FieldsMathematics
InstitutionsStony Brook University
Jon Folkman
John Mather
Laurent C. Siebenmann
Michael Spivak

## Early life and career

Milnor was born on February 20, 1931 in Orange, New Jersey.[2] His father was J. Willard Milnor and his mother was Emily Cox Milnor.[3][4] As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fary–Milnor theorem. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Robert H. Fox.[5] He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox.[6] His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure. Upon completing his doctorate, he went on to work at Princeton. He was a professor at the Institute for Advanced Study from 1970 to 1990.

His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, and Michael Spivak. His wife, Dusa McDuff, is a professor of mathematics at Barnard College.

## Research

One of his published works 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 one considers orientation).

An n-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. He gave a complete inventory of differentiable structures in spheres of all dimensions with Kervaire, and only continued till 2009.

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 fiber has the homotopy type of a bouquet of μ spheres where μ is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area that continues to mature to this day.

In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct.[7][8]

In 1966 the following conjecture on complete surfaces in ${\displaystyle {\mathbb {R} }^{3}}$  was attributed to Milnor: [9]

For any complete, umbilic-free surface with principal curvatures ${\displaystyle \kappa _{1},\kappa _{2}}$ : if the quantity ${\displaystyle \kappa _{1}^{2}+\kappa _{2}^{2}}$  is bounded away from zero, then either the Gauss curvature ${\displaystyle K=\kappa _{1}\kappa _{2}}$  changes sign, or else must vanish identically.

Here an umbilical point on a surface is one where ${\displaystyle \kappa _{1}=\kappa _{2}}$ .

The counter-positive statement splits naturally into two cases: ${\displaystyle K\geq 0}$  and ${\displaystyle K\leq 0}$ . The conjecture is true if one has strict inequalities: strict convexity means that the surface is closed, genus zero and must therefore have umbilical points by the Poincare-Hopf theorem. The strictly negative case, is a celebrated result of Efimov [10] [11]

The full conjecture remains open, although various cases have been proven. When the surface is homeomorphic to a convex plane and one also excludes umbilical points at infinity by demanding ${\displaystyle \inf |\kappa _{2}-\kappa _{1}|>0}$ , Victor Andreevich Toponogov showed that the conjecture holds when either the integral of the Gauss curvature is less than ${\displaystyle 2\pi }$ , or the Gauss curvature and the gradients of the curvatures are bounded. [12]

Fontenele and Xavier proved the convex case of the conjecture when the second fundamental of the surface is bounded below and its gradient is bounded above. [13] Generalisations of the conjecture have been considered in higher dimensions with Ricci curvature replacing Gauss curvature [14] and in the other 3-dimensional constant curvature geometries.[15]

Milnor introduced the growth invariant in a finitely presented group and the theorem stating that the fundamental group of a negatively curved Riemannian manifold has exponential growth became a striking point in the foundation for modern geometric group theory, and the basis for the theory of a hyperbolic group in 1987 by Mikhail Gromov.

In 1984 Milnor introduced a definition of attractor.[16] The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors.

Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of Topological Methods in Modern Mathematics:

It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.[17]

His other significant contributions include microbundles, influencing the usage of Hopf algebras, algebraic K-theory, etc. He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books. He served as Vice President of the AMS in 1976–77 period.

## Awards and honors

Milnor was elected as a member of the American Academy of Arts and Sciences in 1961.[18] In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He later went on to win the National Medal of Science (1967), the Lester R. Ford Award in 1970[19] and again in 1984,[20] the Leroy P. Steele Prize for "Seminal Contribution to Research" (1982), the Wolf Prize in Mathematics (1989), the Leroy P. Steele Prize for Mathematical Exposition (2004), and the Leroy P. Steele Prize for Lifetime Achievement (2011) "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399–405".[21] In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.[22]

Milnor was awarded the 2011 Abel Prize,[23] for his "pioneering discoveries in topology, geometry and algebra."[24] Reacting to the award, Milnor told the New Scientist "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning."[25] In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".[26]

## Publications

### Books

• Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9.[27]
• —— (1965). Lectures on the h-cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324.
• —— (1968). Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN 0-691-08065-8.
• —— (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4.
• Husemoller, Dale; Milnor, John W. (1973). Symmetric bilinear forms. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5.
• Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN 0-691-08122-0.[28]
• Milnor, John W. (1997) [1965]. Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9.
• —— (1999). Dynamics in one complex variable. Wiesbaden, Germany: Vieweg. ISBN 3-528-13130-6.2nd edn. 2000.[29]

## References

1. ^ Milnor's Theorem – from Wolfram MathWorld
2. ^ Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930–1980, p. 35. Institute for Advanced Study, 1980. Accessed November 24, 2015. "Milnor, John Willard M, Topology Born 1931 Orange, NJ."
3. ^ Helge Holden; Ragni Piene (3 February 2014). The Abel Prize 2008–2012. Springer Berlin Heidelberg. pp. 353–360. ISBN 978-3-642-39448-5.
4. ^ Allen G. Debus (1968). World Who's who in Science: A Biographical Dictionary of Notable Scientists from Antiquity to the Present. Marquis-Who's Who. p. 1187.
5. ^ Milnor, John W. (1951). Link groups. Princeton, NJ: Department of Mathematics.
6. ^ Milnor, John W. (1954). Isotopy of links. Princeton, NJ: Department of Mathematics.
7. ^ : –. Cite journal requires |journal= (help); Missing or empty |title= (help)
8. ^ : –. Cite journal requires |journal= (help); Missing or empty |title= (help)
9. ^ Klotz, T.; Osserman, R. (1966). "Complete surfaces in E3 with constant mean curvature". Commentarii Mathematici Helvetici. 41 (1): 313–318. doi:10.1007/BF02566886.
10. ^ Efimov, N.V. (1964). "Generation of singularities on surfaces of negative curvature (in Russian)". Mat. Sbornik. 64(106) (2): 286–320.
11. ^ Efimov, N.V. (1966). "Hyperbolic problems in the theory of surfaces (in Russian)". Proc. Int. Congr. Math. Moscow: 177–188.
12. ^ Toponogov, V.A. (1995). "On conditions for existence of umbilical points on a convex surface". Siberian Mathematical Journal. 36 (4): 780–784. doi:10.1007/BF02107335.
13. ^ Fontenele, F.; Xavier, F. (2019). "Finding umbilics on open convex surfaces". Rev. Mat. Iberoam. 35 (7): 2035–2052.
14. ^ Smyth, B.; Xavier, F. (1987). "Efimov's theorem in dimension greater than two". Inventiones mathematicae. 90 (3): 443–450. doi:10.1007/BF01389174.
15. ^ Gálvez, J.A.; Martínez, A.; Teruel, J.L. (2015). "Complete surfaces with non-positive extrinsic curvature in H³ and S³". Journal of Mathematical Analysis and Applications. 430 (2): 1058–1064. doi:10.1016/j.jmaa.2015.05.049.
16. ^ Milnor, John (1985). "On the concept of attractor". Communications in Mathematical Physics. 99 (2): 177–195. Bibcode:1985CMaPh..99..177M. doi:10.1007/BF01212280. ISSN 0010-3616.
17. ^ Lyubich, Mikhail (1993). Michael Yampolsky (ed.). Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor's 75th Birthday. Houston, Texas. pp. 85–92.
18. ^ "John Willard Milnor". American Academy of Arts & Sciences. Retrieved 2020-05-31.
19. ^ Milnor, John (1969). "A problem in cartography". Amer. Math. Monthly. 76 (10): 1101–1112. doi:10.2307/2317182. JSTOR 2317182.
20. ^ Milnor, John (1983). "On the geometry of the Kepler problem". Amer. Math. Monthly. 90 (6): 353–365. doi:10.2307/2975570. JSTOR 2975570.
21. ^ O'Connor, J J; EF Robertson. "John Willard Milnor".
22. ^ Goldberg, Lisa R.; Phillips, Anthony V., eds. (1993), Topological methods in modern mathematics, Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991, Houston, TX: Publish-or-Perish Press, ISBN 978-0-914098-26-3
23. ^ Abelprisen (Abel Prize) website. "The Abel Prize awarded to John Milnor, Stony Brook University, NY". Archived from the original on April 29, 2011. Retrieved March 24, 2011.
24. ^ Ramachandran, R. (March 24, 2011). "Abel Prize awarded to John Willard Milnor". The Hindu. Retrieved 24 March 2011.
25. ^ Aron, Jacob (March 23, 2011). "Exotic sphere discoverer wins mathematical 'Nobel'". New Scientist. Retrieved 24 March 2011.
26. ^ 2014 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2013-11-04.
27. ^ Kuiper, N. H. (1965). "Review: Morse theory, by John Milnor". Bull. Amer. Math. Soc. 71 (1): 136–137. doi:10.1090/s0002-9904-1965-11251-4.
28. ^ Spanier, E. H. (1975). "Review: Characteristic classes, by John Milnor and James D. Stasheff". Bull. Amer. Math. Soc. 81 (5): 862–866. doi:10.1090/s0002-9904-1975-13864-x.
29. ^ Hubbard, John (2001). "Review: Dynamics in one complex variable, by John Milnor". Bull. Amer. Math. Soc. (N.S.). 38 (4): 495–498. doi:10.1090/s0273-0979-01-00918-1.