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{{Use mdy dates|date=November 2022}}{{Short description|American mathematician}}{{For|those of a similar name|John Milner (disambiguation)}}

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| birth_place = Orange, New Jersey| death_date = | death_place = | nationality = American| field = MathematicsPrinceton University,State University of New York at Stony Brook>Stony Brook UniversityPrinceton University (Bachelor of Arts>AB, PhD)| doctoral_advisor = Ralph FoxTadatoshi Akiba Jon Folkman John N. Mather>John MatherLaurent C. SiebenmannMichael SpivakExotic spheresFáry–Milnor theoremHauptvermutungMilnor K-theoryMicrobundle Milnor Map, Milnor number and Milnor fibration in the theory of complex hypersurface singularities, part of singularity theory and algebraic geometryMilnor–Thurston kneading theoryPlumbing (mathematics)>PlumbingMilnor–Wood inequalitySurgery theoryRokhlin’s theoremSpectral geometry>Isospectral Non-Isometric compact Riemannian manifoldsÅ varc–Milnor lemma| spouse = Dusa McDuffPutnam Fellow (1949, 1950) Sloan Fellowship (1955) Fields Medal (1962) National Medal of Science (1967) Leroy P. Steele Prize (1982, 2004, 2011) Wolf Prize in Mathematics>Wolf Prize (1989) Abel Prize (2011)| religion = | footnotes = }}John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes.

Early life and career

Milnor was born on February 20, 1931, in Orange, New Jersey.Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930–1980 {{Webarchive|url=https://web.archive.org/web/20111124234612library.ias.edu/files/pdfs/hs/cos.pdf#page=35 |date=November 24, 2011 }}, p. 35. Institute for Advanced Study, 1980. Accessed November 24, 2015. “Milnor, John Willard M, Topology Born 1931 Orange, NJ.” His father was J. Willard Milnor, an engineer,WEB, John Milnor - Biography,mathshistory.st-andrews.ac.uk/Biographies/Milnor/, 2023-03-27, Maths History, en, and his mother was Emily Cox Milnor.BOOK, Helge Holden, Ragni Piene, The Abel Prize 2008–2012,books.google.com/books?id=tEprnQEACAAJ, February 3, 2014, Springer Berlin Heidelberg, 978-3-642-39448-5, 353–360, BOOK, Allen G. Debus, World Who’s who in Science: A Biographical Dictionary of Notable Scientists from Antiquity to the Present,books.google.com/books?id=CZNNAQAAIAAJ, 1968, Marquis-Who’s Who, 1187, As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950WEB, Putnam Competition Individual and Team Winners,www.maa.org/programs/maa-awards/putnam-competition-individual-and-team-winners, Mathematical Association of America, December 10, 2021, and also proved the Fáry–Milnor theorem when he was only 19 years old. 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.BOOK, Milnor, John W.,catalog.princeton.edu/catalog/4122388, Link groups, 1951, Princeton, NJ: Department of Mathematics, en, 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.BOOK, Milnor, John W.,catalog.princeton.edu/catalog/1682505, Isotopy of links, 1954, Princeton, NJ: Department of Mathematics, en, His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure, classifying Brunnian links up to link-homotopy and introduced new invariants of it, called Milnor invariants. 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.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 Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, Michael Spivak, and Jonathan Sondow. His wife, Dusa McDuff, is a professor of mathematics at Barnard College and is known for her work in symplectic topology.

Research

One of Milnor’s best-known works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the term exotic sphere, referring to any n-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres, showing in particular that the 7-sphere has 15 distinct differentiable structures (28 if one considers orientation). 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, Singular Points of Complex Hypersurfaces, 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, 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. {{cn|date=December 2020}}In 1984 Milnor introduced a definition of attractor.JOURNAL, Milnor, John, 1985, On the concept of attractor,projecteuclid.org/euclid.cmp/1103942677, Communications in Mathematical Physics, EN, 99, 2, 177–195, 0010-3616, 10.1007/BF01212280, 1985CMaPh..99..177M, 120688149, 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.BOOK, Lyubich, Mikhail, Back to the origin: Milnor’s program in dynamics, Goldberg, Lisa R., Phillips, Anthony Valiant, Topological Methods in Modern Mathematics: A Symposium in Honor of John Milnor’s Sixtieth Birthday, Publish or Perish, 1993, 0-914098-26-8, 85–92, His other significant contributions include microbundles, influencing the usage of Hopf algebras, theory of quadratic forms and the related area of symmetric bilinear forms, higher algebraic K-theory, game theory, and three-dimensional Lie groups.

Awards and honors

Milnor was elected as a member of the American Academy of Arts and Sciences in 1961.WEB, John Willard Milnor,www.amacad.org/person/john-willard-milnor, American Academy of Arts & Sciences, en, May 31, 2020, In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He was elected to the United States National Academy of Sciences in 1963 and the American Philosophical Society 1965.WEB, John W. Milnor,www.nasonline.org/member-directory/members/52491.html, October 6, 2022, www.nasonline.org, WEB, APS Member History,search.amphilsoc.org/memhist/search?creator=John+Milnor&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced, October 6, 2022, search.amphilsoc.org, He later went on to win the National Medal of Science (1967), the Lester R. Ford Award in 1970JOURNAL, Milnor, John, A problem in cartography, Amer. Math. Monthly, 76, 10, 1969, 1101–1112,www.maa.org/programs/maa-awards/writing-awards/a-problem-in-cartography, 10.2307/2317182, 2317182, and again in 1984,JOURNAL, Milnor, John, On the geometry of the Kepler problem, Amer. Math. Monthly, 90, 6, 1983, 353–365,www.maa.org/programs/maa-awards/writing-awards/on-the-geometry-of-the-kepler-problem, 10.2307/2975570, 2975570, 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). In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.{{citation|title=Topological methods in modern mathematics| series=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|editor1-first=Lisa R.|editor1-last=Goldberg|editor2-first=Anthony V.|editor2-last=Phillips|publisher=Publish-or-Perish Press| location=Houston, TX|year=1993|isbn=978-0-914098-26-3}}Milnor was awarded the 2011 Abel Prize,WEB, 2011: John Milnor,abelprize.no/abel-prize-laureates/2011, August 22, 2022, Abelprisen (Abel Prize) website, for his “pioneering discoveries in topology, geometry and algebra.“NEWS, Ramachandran, R., Abel Prize awarded to John Willard Milnor,www.thehindu.com/sci-tech/science/article1565424.ece, March 24, 2011, The Hindu, March 24, 2011, 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.“NEWS, Aron, Jacob, Exotic sphere discoverer wins mathematical ‘Nobel’,www.newscientist.com/article/dn20281-exotic-sphere-discoverer-wins-mathematical-nobel.html, March 24, 2011, New Scientist, March 23, 2011, In 2013 he became a fellow of the American Mathematical Society, for “contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems”.2014 Class of the Fellows of the AMS, American Mathematical Society, retrieved November 4, 2013.In 2020 he received the Lomonosov Gold Medal of the Russian Academy of Sciences.Lomonosov Gold Medal 2020.

Publications

Books

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Journal articles

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• JOURNAL, Michel A., Kervaire, Michel Kervaire, John W., Milnor, John Milnor,www.uni-math.gwdg.de/schick/publ/Groups%20of%20homotopy%20spheres%20I.pdf, Groups of homotopy spheres: I, Annals of Mathematics, Princeton University Press, 77, 1963, 3, 504–537, 10.2307/1970128, 1970128, 0148075,
• JOURNAL, John W., Milnor, Differential topology forty-six years later, Notices of the American Mathematical Society, 58, 2011, 6, 804–809,www.ams.org/notices/201106/rtx110600804p.pdf,

Lecture notes

• BOOK, John Willard, Milnor, James Raymond, Munkres, James Munkres, Lectures on Differential Topology, Collected papers of John Milnor, Volume 4, John Willard, Milnor, American Mathematical Society, 145–176, 2007, 978-0-8218-4230-0,books.google.com/books?id=PrZANUa0dtgC&pg=PA145,

See also

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

References

{{reflist|30em}}

External links

• {{MacTutor Biography|id=Milnor}}
• Home page at SUNYSB
• Photo
• Exotic spheres home page
• www.abelprisen.no/en/multimedia/2011/index.html" title="web.archive.org/web/20120119102126www.abelprisen.no/en/multimedia/2011/index.html">The Abel Prize 2011 – video
• JOURNAL, Raussen, Martin, Skau, Christian, March 2012, Interview with John Milnor, Notices of the American Mathematical Society, 59, 3, 400–408,www.ams.org/notices/201203/rtx120300400p.pdf, 10.1090/noti803, free,
• WEB, Seminar Videos, IMS Video Collection, Institute for Mathematical Sciences, Stony Brook University,www.math.stonybrook.edu/ims-video-collection, (40 links from 1965 to May 2021, with 9 videos from Milnor’s seminars)

{{Fields medalists}}{{Wolf Prize in Mathematics}}{{Abel Prize laureates}}{{Winners of the National Medal of Science|math-stat-comp}}{{Authority control}}