John Milnor

John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in

He was an editor of the Annals of Mathematics for a number of years after 1962. He has written a number of books which are famous for their clarity, presentation, and an inspiration for the research by many mathematicians in their areas even after many decades since their publication. He served as Vice President of the AMS in 1976–77 period.

His students have included

Research

One of Milnor's best-known works is his proof in 1956 of the existence of

In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct, using the concept of Reidemeister torsion. This led to a wave of advances in topology by Milnor and many other mathematicians which changed the perception of the field forever. ^{[citation needed]}

In 1984 Milnor introduced a definition of attractor.^[8] 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.^[9]

His other significant contributions include

Milnor was awarded the 2011 Abel Prize,^[16] for his "pioneering discoveries in topology, geometry and algebra."^[17] 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."^[18]

In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".^[19]

In 2020 he received the Lomonosov Gold Medal of the Russian Academy of Sciences.^[20]

Publications

Books

• Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by
  ISBN 0-691-08008-9.^[21]
• —— (1965). Lectures on the h-cobordism theorem. Notes by
  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
  .
• ISBN 978-0-387-06009-5
  .
• Milnor, John W.;
  ISBN 0-691-08122-0.^[22]
• 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.^[23]

• Milnor, John W. (1956). "On manifolds homeomorphic to the 7-sphere".
  S2CID 18780087
  .
• —— (1959). "Sommes de variétés différentiables et structures différentiables des sphères".
  MR 0117744
  .
• —— (1959b). "Differentiable structures on spheres".
  MR 0110107
  .
• —— (1961). "Two complexes which are homeomorphic but combinatorially distinct". Annals of Mathematics. 74 (2). Princeton University Press: 575–590.
  MR 0133127
  .
• —— (1984). "On the concept of attractor". Communications in Mathematical Physics. 99 (2). Springer Press: 177–195.
  S2CID 120688149
  .
• MR 0148075
  .
• Milnor, John W. (2011). "Differential topology forty-six years later" (PDF). Notices of the American Mathematical Society. 58 (6): 804–809.

Lecture notes

• Milnor, John Willard;
  ISBN 978-0-8218-4230-0
  .

See also

• List of things named after John Milnor
• Orbit portrait
• Microbundle

References

1. ^ Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930–1980 Archived November 24, 2011, at the Wayback Machine, p. 35. Institute for Advanced Study, 1980. Accessed November 24, 2015. "Milnor, John Willard M, Topology Born 1931 Orange, NJ."
2. ^ "John Milnor - Biography". Maths History. Retrieved March 27, 2023.
3. 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. ^ "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 10, 2021.
6. ^ Milnor, John W. (1951). Link groups. Princeton, NJ: Department of Mathematics.
7. ^ Milnor, John W. (1954). Isotopy of links. Princeton, NJ: Department of Mathematics.
8. S2CID 120688149
   .
9. ISBN 0-914098-26-8
   .
10. ^ "John Willard Milnor". American Academy of Arts & Sciences. Retrieved May 31, 2020.
11. ^ "John W. Milnor". www.nasonline.org. Retrieved October 6, 2022.
12. ^ "APS Member History". search.amphilsoc.org. Retrieved October 6, 2022.
13. JSTOR 2317182
    .
14. JSTOR 2975570
    .
15. ISBN 978-0-914098-26-3
16. ^ "2011: John Milnor". Abelprisen (Abel Prize) website. Retrieved August 22, 2022.
17. ^ Ramachandran, R. (March 24, 2011). "Abel Prize awarded to John Willard Milnor". The Hindu. Retrieved March 24, 2011.
18. ^ Aron, Jacob (March 23, 2011). "Exotic sphere discoverer wins mathematical 'Nobel'". New Scientist. Retrieved March 24, 2011.
19. ^ 2014 Class of the Fellows of the AMS, American Mathematical Society, retrieved November 4, 2013.
20. ^ Lomonosov Gold Medal 2020.
21. doi:10.1090/s0002-9904-1965-11251-4
    .
22. doi:10.1090/s0002-9904-1975-13864-x
    .
23. doi:10.1090/s0273-0979-01-00918-1
    .

External links

Wikiquote has quotations related to John Milnor.

• O'Connor, John J.; Robertson, Edmund F., "John Milnor", MacTutor History of Mathematics Archive, University of St Andrews
• Home page at SUNYSB
• Photo
• Exotic spheres home page
• The Abel Prize 2011 – video
• Raussen, Martin; Skau, Christian (March 2012). "Interview with John Milnor" (PDF).
  doi:10.1090/noti803
  .
• "Seminar Videos, IMS Video Collection". Institute for Mathematical Sciences, Stony Brook University. (40 links from 1965 to May 2021, with 9 videos from Milnor's seminars)