John Willard Milnor (born February 20, 1931) is an American mathematician known for his work indifferential topology,algebraic K-theory and low-dimensional holomorphicdynamical systems. Milnor is a distinguished professor atStony Brook University and the only mathematician to have won theFields Medal, theWolf Prize, theAbel Prize and all threeSteele prizes.
Milnor was born on February 20, 1931, inOrange, New Jersey.[1] His father was J. Willard Milnor, an engineer,[2] and his mother was Emily Cox Milnor.[3][4] As an undergraduate atPrinceton University he was named aPutnam Fellow in 1949 and 1950[5] and also proved theFáry–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 ofRalph Fox.[6] 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.[7] His dissertation concernedlink groups (a generalization of the classical knot group) and their associated link structure, classifyingBrunnian links up to link-homotopy and introduced new invariants of it, calledMilnor invariants. Upon completing his doctorate, he went on to work at Princeton. He was a professor at theInstitute for Advanced Study from 1970 to 1990.
He was an editor of theAnnals 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 theAMS in 1976–77 period.
His students have includedTadatoshi Akiba,Jon Folkman,John Mather,Laurent C. Siebenmann,Michael Spivak, and Jonathan Sondow. His wife,Dusa McDuff, is a professor of mathematics atBarnard College and is known for her work insymplectic topology.
One of Milnor's best-known works is his proof in 1956 of the existence of7-dimensionalspheres with nonstandard differentiable structure, which marked the beginning of a new field – differential topology. He coined the termexotic sphere, referring to anyn-sphere with nonstandard differential structure. Kervaire and Milnor initiated the systematic study of exotic spheres byKervaire–Milnor groups, showing in particular that the 7-sphere has 15 distinctdifferentiable 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 asingular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on thetopology of isolatedsingular points of complex hypersurfaces in general, developing the theory of theMilnor fibration whose fiber has thehomotopy type of a bouquet ofμ spheres whereμ is known as theMilnor 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 theHauptvermutung by illustrating twosimplicial complexes that arehomeomorphic butcombinatorially distinct, using the concept ofReidemeister torsion.[8]
In 1984 Milnor introduced a definition ofattractor.[9] 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 ofTopological 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 withThurston. 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 withPoincaré's work oncircle 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.[10]
His other significant contributions includemicrobundles, influencing the usage ofHopf algebras, theory ofquadratic forms and the related area ofsymmetric bilinear forms, higheralgebraic K-theory,game theory, and three-dimensionalLie groups.
Milnor was elected as a member of theAmerican Academy of Arts and Sciences in 1961.[11] In 1962 Milnor was awarded theFields Medal for his work in differential topology. He was elected to the United StatesNational Academy of Sciences in 1963 and theAmerican Philosophical Society 1965.[12][13] He later went on to win theNational Medal of Science (1967), theLester R. Ford Award in 1970[14] and again in 1984,[15] theLeroy P. Steele Prize for "Seminal Contribution to Research" (1982), theWolf Prize in Mathematics (1989), theLeroy P. Steele Prize for Mathematical Exposition (2004), and theLeroy P. Steele Prize for Lifetime Achievement (2011). In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.[16]
Milnor was awarded the 2011Abel Prize,[17] for his "pioneering discoveries in topology, geometry and algebra."[18] Reacting to the award, Milnor told theNew Scientist "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning."[19]
In 2013 he became afellow of theAmerican Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".[20]
In 2020 he received theLomonosov Gold Medal of the Russian Academy of Sciences.[21]
